Hoàng's degree-sequence conjecture for tough graphs

Let $G$ be a simple graph. Write $c(G)$ for its number of components, and define its toughness by

$$\tau(G)=\min\left\{\frac{|S|}{c(G-S)}:S\subseteq V(G),\ c(G-S)\ge 2\right\}$$

when $G$ is not complete, with $\tau(G)=\infty$ otherwise. The graph is $t$-tough if $\tau(G)\ge t$. Let $d_1\le d_2\le\cdots\le d_n$ be the degree sequence of an $n$-vertex graph $G$.

Hoàng's conjecture. Let $n\ge 3$ and $t\ge 1$ be integers. If $G$ is $t$-tough and, for every $i<\frac n2$, $d_i\le i$ implies $d_{n-i+t}\ge n-i$, then $G$ is Hamiltonian.

This conjecture is a toughness analogue of Chvátal's degree-sequence theorem. Hoàng proved it for $t\le 3$, Hoàng and Robin proved it for $t=4$, and the source states that it has been confirmed for all $t\ge 4$; thus the conjecture is solved.