Stochastic diameter monotonicity conjecture for random graphs with fixed surplus

Let $\boldsymbol{d}$ and $\boldsymbol{d}'$ be degree sequences, let $s( \boldsymbol{d})$ denote surplus, and let $n_b( \boldsymbol{d})$ denote the number of vertices of degree $b$. Let $\boldsymbol{G}$ and $\boldsymbol{G}'$ be uniformly distributed over connected graphs with degree sequences $\boldsymbol{d}$ and $\boldsymbol{d}'$, respectively.

Stochastic diameter monotonicity conjecture. If

$$s( \boldsymbol{d})=s( \boldsymbol{d}'),\qquad n_1( \boldsymbol{d})=n_1( \boldsymbol{d}'),\qquad n_2( \boldsymbol{d})=n_2( \boldsymbol{d}'),\qquad n_b( \boldsymbol{d})=0\quad(b>3),$$

then

$$\operatorname{diam}( \boldsymbol{G}')\preceq_{\operatorname{st}}\operatorname{diam}( \boldsymbol{G}).$$

Here $\preceq_{\operatorname{st}}$ denotes stochastic domination.

This would formalize the heuristic that, at fixed surplus and fixed numbers of degree-one and degree-two vertices, allowing higher degrees reduces diameter; the source notes that it would imply sharper expected-diameter bounds for many positive-surplus ensembles.