Itai–Zehavi conjecture on independent spanning trees

Let $d\geq 1$, let $G$ be a $d$-vertex-connected graph, and let $r$ be a vertex of $G$. A family of spanning trees $T_1,\ldots,T_d$ is said to provide independent paths from $r$ if, for every vertex $v\in V(G)\setminus\\{r\\}$, the paths from $v$ to $r$ in the trees are pairwise internally vertex-disjoint. Itai–Zehavi conjecture. For every such $G$ and $r$, there exists a family of $d$ spanning trees $T_1,\ldots,T_d$ providing independent paths from $r$. This conjecture asks for a global strengthening of Menger's theorem, replacing separately chosen disjoint paths by paths organized into spanning trees. The paper proves the conjecture asymptotically with high probability for suitable Erdős–Rényi and random regular graphs, while the general deterministic statement remains open.