Per-split sub-top coefficient conjecture for Miura-ori flip-graph counts

Let $d\ge5$, let $(a,b)$ satisfy $a+b=d$ and $a\le b$, and let $R_{(a,b)}(m,n)$ be the interior remainder. Denote by $[m^{d-3}]R_{(a,b)}(m,n)$ the coefficient of $m^{d-3}$; by symmetry, this is equivalently the coefficient of $n^{d-3}$. Sub-top coefficient conjecture.

$$[m^{d-3}]R_{(a,b)}(m,n)=\begin{cases}\dfrac{8(1+\delta_{ab})}{(d-3)!}&\text{if }|a-b|\le1,\\[2pt]0&\text{if }|a-b|>1. \end{cases}$$

The conjecture refines the proposed degree bound by specifying the pure sub-top coefficient for every split; the stated vanishing has been checked in the finite cases reported in the source, but a general proof remains open.