Positivity conjecture for the Specht-to-web transition coefficients

Let $T'$ be a tableau with corresponding web $w'$, let $v_{T'}$ be the associated Specht basis vector, and let $\phi$ be the equivariant map into web space with web basis vectors $v_w$, where $w\in\mathcal{W}_n$. Write $w\prec w'$ when $w$ is strictly below $w'$ in the partial order induced by the web graph.

Transition-coefficient positivity conjecture. For every tableau $T'$ with corresponding web $w'$, one has

$$\phi(v_{T'})=w'+\sum_{\substack{w\in\mathcal{W}_n\\ w\prec w'}}c_w^{w'}v_w,$$

where $c_w^{w'}>0$ for every $w$ occurring in the sum.

This incorporates the paper's two introductory conjectures: positivity of all transition-matrix entries and nonvanishing whenever $w\preceq w'$. The preceding theorem proves the relevant vanishing outside the web-graph order, while strict positivity of the remaining coefficients is left open.