Prove the edge-boundary conjecture for regular spanning subgraphs

Prove that for every fixed integer d\geq 3 and every sequence F=F(n) of d-regular graphs on [n] satisfying |\partial_e(\widetilde F)|\geq d+1 for every subgraph \widetilde F\subset F with 3\leq |V(\widetilde F)|\leq n-3, the threshold for containing a spanning copy of F satisfies p_c(F)=(1+o(1))p_e(F).

Background

Theorem 1 establishes p_c(F)=(1+o(1))p_e(F) for broad classes of d-regular graphs under additional conditions controlling edge boundaries, automorphisms, or cyclic structure. The authors state that these conditions are likely not optimal.

The conjecture proposes that the sole expansion condition |\partial_e(\widetilde F)|\geq d+1 for all nontrivial, non-co-large subgraphs should suffice. A hexagonal lattice, after regular completion, is identified as a notable special case motivating the conjecture.

References

We actually believe that the following is true. Let $d\geq 3$ and let $F=F(n)$ be a sequence of $d$-regular graphs on $[n]$, $n\in\mathbb{N}$, such that $|\partial_e(\tilde F)|\geq d+1$, for every $\tilde F\subset F$ with $3\leq|V(\tilde F)|\leq n-3$. Then $p_c(F)=(1+o(1))p_e(F)$.

— Sharp thresholds for spanning regular subgraphs  (2502.14794 - Zhukovskii, 20 Feb 2025) in Conjecture in Section 6, “Remaining challenges”