Vakil's geometric Littlewood–Richardson conjecture for partial flag varieties

Fix the partial flag variety $\mathbb{F}l(a_1,\dots,a_m)$ in $n$-space, let $S$ be its set of Schubert cells, and let $d_0,\dots,d_{\binom n2}$ be the Schubert cells in the chosen degeneration order for $Fl(n)$. For $(\alpha,\beta,d_i)\in S\times S\times\{d_0,\dots,d_{\binom n2}\}$, choose flags $M_{\bullet}$ and $F_{\bullet}$ in relative position $d_i$ and define

$$PV_{\alpha,\beta,d_i}=\overline{\Omega_{\alpha}(M_{\bullet})\cap\Omega_{\beta}(F_{\bullet})}.$$

Vakil's geometric Littlewood–Richardson conjecture. There exists a subset $U\subseteq S\times S\times\{d_0,\dots,d_{\binom n2}\}$ such that: $(\alpha,\beta,d_0)\in U$ for all $\alpha,\beta$; if $(\alpha,\beta,d_{\binom n2})\in U$, then $\alpha=\beta$; and if $(\alpha,\beta,d_i)\in U$ with $i<\binom n2$, then degenerating $M_{\bullet}$ from relative position $d_i$ to $d_{i+1}$ makes $PV_{\alpha,\beta,d_i}$ degenerate to a union of puzzle varieties $PV_{\alpha',\beta',d_{i+1}}$ with $(\alpha',\beta',d_{i+1})\in U$, each with multiplicity $1$. This conjecture, attributed in the source to Vakil's Conjecture 4.9, generalizes the Grassmannian and two-step geometric Littlewood–Richardson rules and was reported there as verified up to $n=5$.