Millions of people marched against climate crisis over the past two weeks, in some of the largest demonstrations of the millennium. Most people cheered the students who led the rallies – call them the Greta Generation. But now we’ll start to find out if all their earnest protest actually matters.

Perhaps the first real test will come on 15 October, when the board of the EU’s European Investment Bank – the largest public bank in the world – meets to decide whether the time has finally come to stop expanding the fossil fuel sector. This should be a no-brainer decision: the bank’s staff has put forward a cogent proposal, supported by campaigners across the continent, that would end loans to new fossil fuel projects by 2020.

That plan fits with the facts: when the world’s climate scientists declared last autumn that we would need to have fundamentally transformed our energy sector within a decade, it was clear that the first job was to stop building any new infrastructure. The first rule of holes is, when you’re in one, stop digging.

In this case that means no more digging for gas pipelines or ports or anything else that will help lock in carbon emissions for decades to come. In the past week of Guardian reporting we’ve learned that the biggest oil companies plan to increase production as much as 35% in the next decade. It’s going to be hard enough to phase out the vast existing fossil fuel infrastructure in the years ahead: adding new projects at this point is insane.

And if the EIB does act, it will send a strong signal to markets and to other lenders. For almost a decade now, observers have understood that restricting the flow of money to the fossil fuel industry is a key part of the climate fight. That’s why endowments and portfolios worth more than $11tn have begun divesting their fossil fuel stocks; last month the University of California system became the latest big player to join in, scrubbing its $80bn endowment and pension fund of fossil fuel stocks. Heck, even a major American utility announced that it was divesting its pension fund because it could see where the future lay.

This kind of effort is beginning to actually pinch – Shell said last year that divestment had become a material risk to its business. Lending to this industry undercuts those efforts.

Ending that lending would also strengthen the growing movement to stop private banks underwriting climate destruction. The big money-centre banks, the kind that lend to governments and businesses, have stepped up their financial support for the fossil fuel industry in the years since the Paris climate agreement, but a growing citizen movement is calling them to account: last week in the US activists closed down branches of Chase Bank, the largest oil and gas lender.

The EIB would also be doing a service to everyone who will be hurt financially as the fossil fuel industry founders. As the governor of the Bank of England, Mark Carney, began warning some years ago, there are massive “stranded assets” represented by oil and gas reserves that can’t be burned if we are to meet our climate targets. This “carbon bubble” hangs over our markets like the housing bubble of a decade ago: if the EIB backs away from new fossil fuel lending, it will help with the slow deflation of that bubble, lessening pressure on the system.

But the biggest reason for the EIB to take this step is simple: it responds to the cries of both young people and scientists.

Researchers have looked on in anguish for many years now, as their warnings have gone largely unheeded – the results so far this year include the hottest month ever recorded on our planet, giant fires across the northern latitudes and a hurricane that stalled over the Bahamas producing what meteorologists called “the longest siege of violent, destructive weather ever observed” on our planet. The iron law of climate crisis is that those who have done the least get hurt the most, but it’s now reaching the stage where even the affluent are at risk: across California last week hundreds of thousands were without power because the state’s largest utility company shut off supplies for fear of sparking forest fires in this hot and arid autumn.

Young people have looked at this data and drawn the obvious conclusion: that we must do far more than we are doing at present. They’ve done the work to understand the crisis that shadows their lives, and they’ve asked adults to join in the fight for a livable future. They’ve also said – loud and clear – that they don’t want to be patted on the head and thanked for providing an inspiring moment. They want action.

Action is what the EIB can provide. Obviously there will be pressure from the oil and gas industry to keep handing over public money: that’s evident in the gas industry loopholes that have fractured the boldness of the initial proposals. What had the potential to be an unmistakable signal to the world that the end of the fossil fuel age was in sight could end up a more muddled message if the EIB bends to industry pressure.

Anyone who believes in physics and chemistry knows that the time for change is here – indeed, it’s past due.

The fifteenth of October is a crucial day in the most important fight the planet has ever faced, and a sign of whether Europe’s governments, particularly those of Germany and Italy, who insiders report have been strongly against a fossil-free EIB policy, will respond with open hearts to the outpouring of hope we have seen in the past weeks.

• Bill McKibben is the founder of the grassroots climate campaign 350.org and author of Falter: Has the Human Game Begun to Play Itself Out?