Uniform perturbation conjecture for motif-support size

Let $\degree$ be a degree sequence, and let $\psi(\degree)$ denote the number of nonzero entries of the associated multigraph representation. Define $v(\degree)$ as the smallest real number such that, for all $i$ and $j$,

$$\left|\psi(\degree+\mathbf{e}_i+\mathbf{e}_j)-\psi(\degree)\right|\leq v(\degree).$$

Let $u(\degree)$ be the analogous perturbation constant for $\beta$. Uniform perturbation conjecture. For all $\degree$, $u(\degree)\leq 1$ and $v(\degree)\leq 1$. The claim asserts uniform stability of both the parameter and the support-size statistic under adding two degree units; the supplied text provides only trivial broad bounds before stating the conjecture.