The two-orientation deletability conjecture for cubic 3-edge-connected graphs

Let $G$ be a cubic $3$-edge-connected graph. An orientation $D$ of $G$ is strongly connected when every vertex is reachable from every other vertex by a directed path. An arc $a$ is deletable from $D$ when $D-a$ is strongly connected.

Two-orientation deletability conjecture. For every cubic $3$-edge-connected graph $G$, there exist two strongly connected orientations $D_1$ and $D_2$ of $G$ such that for every vertex $v$, there exist two arcs $a^1_v$ and $a^2_v$ incident to $v$ such that $D_1-a^1_v$ and $D_2-a^2_v$ are strongly connected.

This is one of the conjectures proposed as a relaxation of the stronger conjecture that every $3$-edge-connected graph has Frank number at most $3$. The source gives no resolution, so it remains open.