Can science provide evidence for supernatural agency?

To be clear, I’m not advocating god-of-the-gaps thinking, and acknowledge the risk of falling into it.

Rather, I’m considering the possibility of situations where scientific evidence might reasonably suggest divine action. A key qualifier is, has the field in question had sufficient time to thoroughly search for naturalistic explanations? Of course, that’s a subjective assessment, and so this will be an “on the balance of the evidence” individual judgement.

Here’s an illustration of what I’m getting at—this approach applied to abiogenesis:

“After 70 years of research since Miller-Urey, the field has had minimal real progress; in fact, the findings increasingly suggest that the problem is intractable. Therefore, a provisional God hypothesis is warranted. Moreover, if research over the next 70 years is similarly limited, this hypothesis would be strengthened.”

I’m not arguing here about this particular example, but using it to show the form of the argument. And clearly with this example, many reject the conclusion of intractability as premature, calling it god-of-the-gaps thinking.

But let’s say that after a further 200 years (or 500, or 1000) of continuous and concerted research, progress remains demonstrably limited. Would the God hypothesis then be a reasonable provisional option?