Arc-disjoint branchings in 2-arc-strong digraphs of independence number 2

Let $D=(V,A)$ be a digraph. Write $\alpha(D)$ for its independence number, let $\lambda(D)$ denote its arc-connectivity, and let $B_s^+$ and $B_s^-$ denote, respectively, an out-branching and an in-branching rooted at $s$. Branching conjecture for independence number 2. Every 2-arc-strong digraph $D$ with $\alpha(D)=2$ has a pair of arc-disjoint branchings $B_s^+,B_s^-$ for every choice of $s\in V$. The source presents this as an open conjecture, alongside a theorem giving a related result for some choice of roots and a 3-arc-strong variant.