The diagram q-Foulkes equivalence conjecture

Let $\alpha,\beta,\gamma,\delta$ be partitions such that

$$e_{\alpha'}[e_{\beta'}]=e_{\gamma'}[e_{\delta'}].$$

Write $\prec_s$ for strict Schur order and let $S_\rho$ denote the corresponding $q$-analogue used in the paper. The diagram q-Foulkes equivalence conjecture.

$$s_\alpha[s_\beta]\prec_s s_\gamma[s_\delta] \qquad\text{if and only if}\qquad 0\prec_s\frac{S_\gamma[S_\delta]-S_\alpha[S_\beta]}{1-q}.$$

This proposed equivalence would reduce the diagrammatic $q$-versions to the $q=0$ Schur-positivity condition. It was checked computationally for the configurations considered in the paper, but the general assertion remains open.