Russell–Tymoczko positivity conjecture for the transitioning matrix

Let $\mathcal{T}(n,n)$ be the set of standard Young tableaux of shape $(n,n)$, let $\{v_T:T\in\mathcal{T}(n,n)\}$ and $\{w_S:S\in\mathcal{T}(n,n)\}$ be the tableau and web bases of the Specht module, and write

$$\rho(v_T)=\sum_{S\in\mathcal{T}(n,n)}a_{ST}w_S.$$

Let $\leq$ be the partial order on $\mathcal{T}(n,n)$ defined by directed paths in the tableaux graph. Russell–Tymoczko's positivity conjecture. The entries satisfy $a_{ST}>0$ if and only if $S\leq T$. This refines the known upper-triangularity of the transitioning matrix with ones on the diagonal; the claim concerns precisely which entries are nonzero and positive, and its resolution is not established by the supplied text.