Resilience conjecture for universality of bounded-degree spanning trees

Fix $\Delta\ge 2$ and $\gamma>0$. Let $\mathcal{T}(n,\Delta)$ denote the family of spanning trees on $n$ vertices with maximum degree at most $\Delta$. The local resilience of a graph with respect to a property is the largest number of incident edges that may be deleted at every vertex while the property is retained.

Bounded-degree tree universality resilience conjecture. The conclusion of the cited resilience theorem—that the local resilience of $G(n,p)$ with respect to being universal for $\mathcal{T}(n,\Delta)$ is almost surely at least $\frac12-\gamma$—holds for $p\ge C\log n/n$.

The source says this probability is conjectured to be optimal up to constants, while the stated theorem only establishes the conclusion for $p\ge C(\log n/n)^{1/3}$.