The stronger Ban–Linial conjecture for orientable bisections

Let $G$ be a cubic graph that admits a perfect matching. An orientable $4$-weak bisection is a $4$-weak bisection with the orientability property used in the source; a $(3\frac{1}{3},\frac{1}{3})$-flow is the corresponding flow notion. Stronger Ban–Linial conjecture. Every cubic graph $G$ that admits a perfect matching, other than the Petersen graph, admits an orientable $4$-weak bisection and, equivalently, a $(3\frac{1}{3},\frac{1}{3})$-flow. This is proposed as a strengthening of the revised Ban–Linial conjecture; the source does not state whether it has been resolved.