The specific character–web conjecture for $\mathfrak{sl}_3$ Grassmannians

Let $T\in\operatorname{SSYT}(3,[m])$ be a semistandard Young tableau, let $\operatorname{ch}(T)$ be its character element in $\mathbb{C}[\operatorname{Gr}(3,m)]$, and let $[W(T)]$ be the web invariant labeled by $T$.

Character–web conjecture. If $\operatorname{ch}(T)$ is a cluster monomial, then

$$\operatorname{ch}(T)=[W(T)].$$

This is the specific version stated after the general web-invariant conjecture, which the source notes is known for the finite mutation type cases $m\leq 9$; no general resolution is supplied.