Extremal transition conjecture for quantum cohomology of toric degenerations

Let $X_{\rm sm}=\operatorname{Fl}(n_1,\dots,n_l,n)$, $X_{\rm sing}$, and $X_{\rm res}$ be as above. Write $q_1,\dots,q_l$ for the Novikov variables associated with the curve classes $C_1,\dots,C_l$, and $q_b$ for those associated with $C_b$, $b\in\operatorname{Box}$. Let $N_b$ denote the residue endomorphisms along $q_b=1$, and define

$$V=\bigcap_{b\in\operatorname{Box}}\operatorname{Ker}N_b,\qquad W=V\cap\sum_{b\in\operatorname{Box}}\operatorname{Im}N_b.$$

Extremal transition conjecture. The following hold: (1) the structure constants of the small quantum product of $X_{\rm res}$ are polynomials in $q_1,\dots,q_l$ with coefficients rational in the $q_b$; (2) its small quantum connection has logarithmic singularities along $\prod_{b\in\operatorname{Box}}(q_b-1)=0$, with nilpotent residue endomorphisms, and the residues $N_b$ are independent of $q_1,\dots,q_l$; (3) along $\Delta_{\rm exc}=\{q_b=1\ (\forall b\in\operatorname{Box})\}$, the induced residual flat connection on $(V/W)\times\Delta_{\rm exc}$ is intertwined by a pairing-preserving linear map $\theta\colon V/W\to H^*(X_{\rm sm})$ with the small quantum connection of $X_{\rm sm}$ under $q_i=\overline q_i$; and (4), for the resolution $\pi\colon X_{\rm res}\to X_{\rm sing}$ and retraction $r\colon X_{\rm sm}\to X_{\rm sing}$, one has $\operatorname{Im}\pi^*\subset V$ and the stated diagram involving $V$, $V/W$, $H^*(X_{\rm sm})$, and $H^*(X_{\rm sing})$ commutes. This is a detailed prediction that the quantum cohomology and quantum connection of the resolution specialize across the extremal transition to those of the smoothing, while describing the exceptional-direction monodromy; the parser supplies no evidence of resolution, so its status remains open.