The x not 1 modulo k−1 conjecture for prographs and set-valued tableaux

Assume $x\equiv a\pmod{k-1}$, where $2\leq a\leq k-1$. Let

$$\lambda=\left(n+\frac{x+k-a-1}{k-1},n+\frac{x+k-a-1}{k-1},m\right),\qquad \mu=\left(\frac{x+k-a-1}{k-1},0,0\right),$$

and let $\operatorname{PC}_x^k(n,m)$ denote the relevant prographs and $\operatorname{\mathbb{S}}(\lambda/\mu,\rho)$ the corresponding set-valued Young tableaux. For $T\in\operatorname{\mathbb{S}}(\lambda/\mu,\rho)$, write $b_1<b_2<\cdots$ for its middle-row entries and $c_1<c_2<\cdots$ for its bottom-row entries. The x not 1 modulo $k-1$ conjecture. The set $\operatorname{PC}_x^k(n,m)$ is in bijection with the subset of tableaux in $\operatorname{\mathbb{S}}(\lambda/\mu,\rho)$ satisfying

$$b_i=i\quad\text{for all }1\leq i\leq k-a,$$

and

$$c_i>b_{(k-1)i+2-(k-a)}\quad\text{for all }1\leq i\leq m-1.$$

The first condition excludes tableaux corresponding to the $k-a$ leftmost children of the initial coproduct terminating at a coproduct node; the second excludes those edges serving as inputs to a product other than the final product. The proposed bijection refines the general prograph–tableau correspondence, but the source gives no resolution of the conjecture.