Conjecture on optimal independent spanning trees in random regular graphs

Let $d=d(n)\in[3,n-1]$, and let $G\sim G(n,d)$ be a random $d$-regular graph. An independent spanning tree (IST) family is a collection of spanning trees whose root-to-vertex paths are pairwise internally vertex-disjoint. Conjecture on optimal independent spanning trees in random regular graphs. With high probability, $G$ contains $d$ ISTs if $d\neq n-3$, and contains $d-1$ ISTs if $d=n-3$. The paper identifies this as a strengthening of the sparse random-regular-graph question; the conjecture remains open.