Uniform upper bound conjecture for diameters of random graphs with minimum degree

For integers $n$ and $d$, let $\mathcal S_{n,d}$ be the set of degree sequences $\boldsymbol{d}=(d_1,\ldots,d_n)$ with $\min_i d_i\geq d$, and let $\boldsymbol{G}$ be uniformly distributed over graphs with degree sequence $\boldsymbol{d}$, denoted $\boldsymbol{G}\in_u\mathcal G_{ \boldsymbol{d}}$.

Minimum-degree diameter conjecture. For every $d\geq3$ and $\epsilon>0$,

$$\limsup_{n\to\infty}\sup_{ \boldsymbol{d}\in\mathcal S_{n,d}}\mathbb P_{ \boldsymbol{G}\in_u\mathcal G_{ \boldsymbol{d}}}\left(\operatorname{diam}( \boldsymbol{G})>(1+ \epsilon)\log_{d-1}n\right)=0.$$

The conjecture asserts that random $d$-regular graphs have, to first order, the largest diameter among random graphs whose minimum degree is at least $d$, uniformly over all admissible degree sequences.