Stanley–Reisner-type degeneration conjecture for loop-model schemes

Let $\rho$ be a link pattern, let $\mathcal D$ be a diagram of $\rho$, let $L_\rho$ be the associated scheme, and let $L_{\rho,\pi}$ denote its fixed-connectivity part for a connectivity $\pi$. For every crossing $i<j<\rho(i)<\rho(j)$ of $\rho$, the relevant component is described by the equations $M_{i,j}=0$ and $M_{j,\rho(i)}=0$. Loop-model degeneration conjecture. There exists a (partial) Gröbner $T_\rho$-equivariant degeneration of $L_\rho$ into an in-general-unreduced scheme whose components are indexed by noncrossing loop configurations of $\mathcal D$, with the components coming from $L_{\rho,\pi}$ indexed by loop configurations of connectivity $\pi$; each geometric component is given by the stated equations and has multiplicity $2^{|\text{loops}|}$. The conjecture refines the expected Stanley–Reisner degeneration by allowing nonreduced components, whose multiplicities account for the loop weight. The supplied text gives no resolution, so it remains open.