Product formula for coefficients of multi-t Macdonald polynomials

Prove that if \widetilde{H}_mu(X;q,\mathbf{t})=\sum_lambda \widetilde{K}_{lambda mu}[q,\mathbf{t}]s_lambda and \widetilde{K}[q,\mathbf{t}]=\sum_{k,\alpha}c_{k,\alpha}q^k\mathbf{t}^{\alpha}, then the product of the monomials q^kT^\alpha, each repeated c_{k,\alpha} times, equals \left(\prod_{(i,j)\in\mu}q^{i-1}t_{j-1}\right)^{d_mu}, where d_mu counts standard tableaux of shape mu in which 2 lies above 1 and t_0=1.

Background

The paper studies a multiplicative statistic of the monomials appearing in Schur coefficients of the multi-t Macdonald polynomials. The proposed identity relates this product to the Ferrers-diagram cells of the partition and to the number d_mu of standard tableaux satisfying a specified positional condition.

The authors state that the same property appears to hold for ordinary Macdonald polynomials and for two computed non-Macdonald examples arising from Yang–Baxter traces, but they leave the general assertion as a conjecture.

References

Let $\tilde H_\mu(X;q,{\bf t})=\sum_\lambda \tilde K[q,{\bf t}]s_\lambda$ and write \begin{equation} \tilde K[q,{\bf t}]= \sum_{k,\alpha} c_{k,\alpha} qk {\bf t}\alpha \end{equation} as a sum of monomials. Then, \begin{equation} \prod_{k,\alpha}(qkT\alpha){c_{k,\alpha}}= \left(\prod_{(i,j)\in\mu} q{i-1}t_{j-1}\right){d_\mu} \end{equation} where $(i,j)$ are the cells of the Ferrers diagram of $\mu$ and $d_\mu$ is the number of standard tableaux of shape $\mu$ in which 2 is above 1, and we set $t_0=1$.

— Noncommutative chromatic quasi-symmetric functions, Macdonald polynomials, and the Yang-Baxter equation  (2502.09072 - Novelli et al., 13 Feb 2025) in Conjecture in Section 4.3, following the discussion of normalized Yang–Baxter traces