Degree bound for remainder terms in Miura-ori flip-graph counts

Let $d\ge5$ and let $(a,b)$ be a split with $a\le b$. Write $R_{(a,b)}$ for the remainder after removing the single-side contributions from the count associated with the split. Degree-bound conjecture. For every such split, $R_{(a,b)}$ has total degree at most $d-3$. This would show that the single-side configurations account for the entire total-degree-$(d-2)$ part of the flip-graph count, while the corresponding bound for all $d\ge5$ remains unproved.