The common-bound conjecture for P-stable degree-sequence families

Let $\mathbb{D}$ be a P-stable family of degree sequences, or a P-stable simple region, and let $\partial(D)$ denote the boundary of a graphic degree sequence $D\in\mathbb{D}$. Let $|D|$ denote the size of $D$. Common-bound conjecture. There is a polynomial $p^*(n)$ such that, for each such P-stable family $\mathbb{D}$, or just for each P-stable simple region $\mathbb{D}$,

$$\partial(D)\le p^*(|D|)$$

for all but finitely many graphic $D\in\mathbb{D}$. The conjecture proposes a uniform polynomial bound on the boundary quantity for P-stable regions; the surrounding discussion notes empirical evidence for the candidate polynomial $p_0(n)=n^{10}$, but does not establish the conjecture.