Jordan-type conjecture for affine growth diagrams

Let $Fl_n$ be the type $A$ affine flag variety of chains of lattices $F=(\cdots\subset F_i\subset\cdots)$, with $F_{i+n}=tF_i$ and one-dimensional successive quotients. For two affine flags $F,F'\in Fl_n$, let $w=w(F,F')$ be their relative position, and define

$$V_{ij}=F_i/(F_i\cap F'_j).$$

For $er\in\mathfrak{n}_F\cap\mathfrak{n}_{F'}$, where $\mathfrak{n}_F$ is the pro-nilpotent radical of the Iwahori Lie algebra stabilizing $F$, the restriction $er|_{V_{ij}}$ is nilpotent and has a Jordan canonical form.

Jordan-type conjecture. There is an open set $\mathcal{N}\subseteq\mathfrak{n}_F\cap\mathfrak{n}_{F'}$ such that, for every $er\in\mathcal{N}$,

$$\operatorname{JCF}(er|_{V_{ij}})=\Gamma_w(i,j),$$

where $\Gamma_w(i,j)$ is the partition at position $(i,j)$ in the growth diagram of $w$.

This conjecture gives the affine growth diagram a geometric interpretation through generic Jordan types of restrictions of topologically nilpotent operators. It complements the combinatorial dual affine Robinson–Schensted correspondence; the source does not report a resolution.