Conjecture on optimal independent spanning trees in Erdős–Rényi graphs

Let $G\sim G(n,p)$ be an Erdős–Rényi random graph, and write $\delta(G)$ for its minimum degree. A spanning tree rooted at $r$ is an independent spanning tree (IST) family member when the paths from every vertex to $r$ in the different trees are pairwise internally vertex-disjoint. Conjecture on optimal independent spanning trees. Fix any $p\geq \log n/n$. Then, with high probability, for every vertex $r\in V(G)$, the graph $G$ contains $\delta(G)$ ISTs rooted at $r$. This would extend the paper's asymptotic result to the threshold regime and attain the natural upper bound supplied by vertex connectivity, which equals the minimum degree in $G(n,p)$ with high probability. The conjecture is posed as an open question.