In his summary of his book Bounded gaps between primes: the epic breakthroughs of the early 21st century, Kevin Broughan writes
Which brings me to my final remark: where to next in the bounded gaps saga? As hinted before, the structure of narrow admissible tuples related to the structure of multiple divisors of Maynard/Tao, and variations of the perturbation structure of Polymath8b, and of the polynomial basis used in the optimization step, could assist progress to the next target. Based on “jumping champions” results, this should be 210. But who knows! [emphasis added]
We have a few years of computer and also presumably analytic development since Polymath8, are we able to push the bound down to 210 with current technology? What would it take, if not just someone with enough motivation to sit down and do it/spend the computational cycles? Are we still a long way from getting to below 200?
These are of course arbitrary numbers, in the grand scheme of things, since true progress down towards the bound of 12 enabled by the Elliott–Halberstam conjecture requires absolutely new ideas that seem too far from current knowledge. But there should be satisfaction in making some progress, even if not fundamentally substantial, similar to the recent case-by-case progress by Booker and Sutherland on outstanding sum-of-three-cubes cases below 1000.