The asymptotic containment conjecture for tuples of standard Young tableaux

Let $A_1,\ldots,A_k$ be standard Young tableaux with $\operatorname{sh}(A_i)=\alpha_i\vdash a_i$ for $i\in[k]$. Let $\mathcal{T}_n(A_1,\ldots,A_k)$ be the set of $k$-tuples $(T_1,\ldots,T_k)$ such that

$$\operatorname{sh}(T_1)=\cdots=\operatorname{sh}(T_k)$$

and $T_i\in\mathcal{T}_n(A_i)$ for all $i\in[k]$. Write $E_1=\cdots=E_k=\emptyset$. Asymptotic containment conjecture.

$$\lim_{n\to\infty}\frac{|\mathcal{T}_n(A_1,\ldots,A_k)|}{|\mathcal{T}_n(E_1,\ldots,E_k)|}=\frac{f^{\alpha_1}\cdots f^{\alpha_k}}{a_1!\cdots a_k!}.$$

This generalizes the preceding one- and two-tableau asymptotic formulas: imposing the condition that all tableaux have the same shape is conjectured not to alter the limiting probability of simultaneous containment.