The independence-number-two conjecture for prescribed roots

Let $D=(V,A)$ be a digraph, and let $calpha(D)$ denote its independence number. An out-branching rooted at $s$ is denoted by $B_s^+$, and an in-branching rooted at $t$ is denoted by $B_t^-$. The branchings are arc-disjoint if they have no common arcs. A digraph is $3$-arc-strong if its arc-connectivity is at least $3$.

Prescribed-roots conjecture. Every $3$-arc-strong digraph $D=(V,A)$ with $calpha(D)=2$ has a pair of arc-disjoint branchings $B_s^+,B_t^-$ for every choice of $s,t\ro V$.

This strengthens the preceding independence-number-two claim by requiring arbitrary, possibly different, roots and is presented as an open conjecture in the source.