Boundary-correction structure conjecture for Miura-ori flip-graph counts

Let $E_d(m,n)$ be the count and let $p_d(m,n)$ be the closed-form polynomial. When $s=\min(m,n)<d-1$, write the correction in the larger variable as

$$B_d^{(s)}(n):=E_d(s,n)-p_d(s,n).$$

Boundary-correction structure conjecture. For every $d\ge4$ and every $m,n$ with exactly one of them below $d-1$, the count $E_d(m,n)$ differs from $p_d(m,n)$ by a polynomial correction in the larger variable, determined by $\min(m,n)$. The source records this polynomial behavior on the threshold range and gives explicit examples; the asserted uniform structure beyond those examples remains open.