Nonzero trace for permutation weights and degree of Tr RSK_{m,n,d}

Prove that Tr RSK_{1^d,1^d} ≠ 0 for all integers d ≥ 2; equivalently, establish that the trace Tr RSK_{m,n,d} has total degree 2d as a polynomial in m and n for all d ≠ 2.

Background

Theorem 8.1 shows that for fixed d, Tr RSK_{m,n,d} is a polynomial whose leading term is determined by Tr RSK_{1^d,1^d}. Computations up to d = 11 suggest varying signs and magnitudes, with Tr RSK_{1^d,1^d} vanishing at d = 2 but seemingly nonzero for other d. The conjecture asserts persistent nonvanishing in the permutation-weight block, giving the expected leading-degree 2d growth for the trace.