Symmetric-extension certificates for the Heo-Kiem range-criterion problem: the cases (7,7,3,9) and (5,14,7,10)

Heo and Kiem (Linear Algebra Appl. 567 (2019) 167-179, Question 1.2) asked whether, for k+l=m+n-2 with an integral zero Pᵏ⁺ˡₘ₋₁(k)=0 of a binary Krawtchouk polynomial, there are subspaces D, E of C^m (x) C^n of codimensions k, l such that no nonzero x, y satisfy x(x)y in D and conj(x)(x)y in E. A previous note (doi:10.5281/zenodo.24132971) settled (5,5,3,5), (6,6,3,7) and (7,7,5,7) with a level-one matrix-inequality certificate. For the pairs found in the cases (7,7,3,9) and (5,14,7,10) no such certificate was found (the semidefinite program has value 0 numerically). We lift the certificate to the symmetric square of the first factor (a level-two Doherty-Parrilo-Spedalieri type certificate) and verify it in exact integer arithmetic (SageMath/FLINT). This answers the question affirmatively for (7,7,3,9) and (5,14,7,10) (and (7,7,9,3), (5,14,10,7)), and gives a second pair for (6,6,3,7). The release contains the note (PDF + LaTeX), the data, the exact verifier (python scripts/verify_dps2.py; numpy + sympy) and the integer matrices with a Sage script. Work done with AI assistance (Anthropic Claude); reviewed by separate AI referees (reports included), not yet by a human expert.