Mixing-time bound for giant components of random graphs with given degrees

Let $m$ be the number of edges, let $n_0(\mathcal D)$ denote the number of vertices of degree zero, let $m^{\neq 2}$ denote the number of edges incident to vertices whose degrees are not equal to $2$, and let $\Delta$ be the maximum degree. A degree sequence is feasible if it is realizable by a graph, and a graph has a giant component if it has a component of linear order. For any function $h$ tending to infinity with $m$, let $(\mathcal D_{\ell})_{\ell\geq 1}$ be a sequence of feasible degree sequences. Mixing-time conjecture. If $n_0(\mathcal D_{\ell})=0$ and $G(\mathcal D_{\ell})$ has a giant component, then with high probability the mixing time of every component is

$$O\left(h(m)(\log m^{\neq 2})^2\max\left(\frac{\Delta^2}{m},\left(\frac{m}{m^{\neq 2}}\right)^2\right)\right).$$

The conjecture would give a near-optimal general upper bound for mixing times while accounting for large degrees and long degree-two chains; the examples preceding it show why restrictions on the degree sequence are necessary. Its resolution status is not specified in the source.