The row-module dual canonical basis conjecture for shifted Yangians

Let $\boldsymbol{\boldsymbol{\text?}}$ be a tensor product parameter with associated free abelian group $S^{\boldsymbol{\boldsymbol{\text?}}}(V_{\mathbb Z})$, and let $k:S^{\boldsymbol{\boldsymbol{\text?}}}(V_{\mathbb Z})\to[\mathcal M_0(\boldsymbol{\boldsymbol{\text?}})]$ be the isomorphism sending $M_A$ to $[M(A)]$ for row tableaux $A$. For each $A$, write $L_A$ for the corresponding dual canonical basis element. The row-module conjecture. The map $k$ should send $L_A$ to the class $[L(A)]$ of the irreducible module $L(A)$; equivalently, for all row tableaux $A,B$, the decomposition number should satisfy

$$[M(A):L(B)]=P_{d(\rho(A))w_0,d(\rho(B))w_0}(1).$$

This is a type-$A$ refinement of the Kazhdan–Lusztig conjecture for shifted Yangians. It is known when the parameter consists of one column and when it has two rows; the general case remains open.