Reiner–Tenner–Yong's barely set-valued tableau enumeration conjecture

A barely set-valued semistandard Young tableau is a set-valued semistandard Young tableau in which exactly one square is assigned two integers and every other square is assigned a single integer. For a partition $\lambda$ and a positive integer $k$, let $\mathrm{BSSYT}(\lambda,k)$ and $\mathrm{SYT}(\lambda,k)$ denote, respectively, the sets of barely set-valued semistandard Young tableaux and ordinary semistandard Young tableaux of shape $\lambda$ such that every integer in row $i$ is at most $k+i$. Let $\delta_d=(d-1,d-2,\ldots,1)$, and let $\delta_d(b^a)$ be the rectangular staircase shape obtained by replacing each square of $\delta_d$ by an $a\times b$ rectangle. Reiner, Tenner and Yong's conjecture. For any positive integers $a,b,d$ and $k$,

$$|\mathrm{BSSYT}(\delta_d(b^a),k)|=\frac{kab(d-1)}{a+b}|\mathrm{SYT}(\delta_d(b^a),k)|.$$

This conjecture concerns the enumeration of barely set-valued tableaux and was posed in the study of the probability distribution of edges in the Young lattice. The paper proves the conjecture, so the assertion is solved.