The nonvanishing conjecture for Schur Eisenstein transition coefficients

Let $\nu\subseteq\lambda$ be partitions, let $d_\lambda(\nu)$ be the corresponding transition coefficient, and let $F_{\lambda\nu}$ be the associated element of the symmetric-function algebra. Let $\Phi_\beta$ be the specialization determined by the renormalized Schur multiple zeta values. Transition-coefficient nonvanishing conjecture. For every pair of partitions $\nu\subseteq\lambda$,

$$d_\lambda(\nu)\neq0,\qquad\text{equivalently}\qquad \Phi_\beta(F_{\lambda\nu})\neq0.$$

For straight shapes, computations suggest that the necessary support condition is sufficient; the conjecture asserts that the full $\beta$-weighted sum does not vanish, even when individual terms in its factorization can vanish.