The dual canonical basis conjecture for the base affine space

Let $k$ be a positive integer, let ${\rm SSYT}(k-1,[k])$ denote the monoid of semistandard tableaux with at most $k-1$ rows and entries in $[k]$, and write each $T\in {\rm SSYT}(k-1,[k])$ as $T=T”\cup T'$, where $T'$ has fundamental tableaux as columns and $T”$ is a fraction of two trivial tableaux. For a tableau $T”$, let $\Delta_{T”}$ be the corresponding product or quotient of flag minors, and define

$${\operatorname{ch}}'(T)=\Delta_{T”}{\operatorname{ch}}_{\widetilde{{\mathbb {C}}[SL_k]^{N^-}}}(T').$$

The dual canonical basis conjecture. For every $T\in {\rm SSYT}(k-1,[k])$,

$${\operatorname{ch}}'(T)\in {\mathbb {C}}[SL_k]^{N^-}.$$

Moreover, the set

$$\{{\operatorname{ch}}'(T):T\in {\rm SSYT}(k-1,[k])\}$$

is the dual canonical basis of ${\mathbb {C}}[SL_k]^{N^-}$. The conjecture asserts that the tableau-indexed formulas, initially defined using the localized algebra $\widetilde{{\mathbb {C}}[SL_k]^{N^-}}$, give regular functions and exhaust the dual canonical basis of the base affine space.