Exact 2-dimensional flow number conjecture for the Petersen graph

Let $P$ be the Petersen graph, and let $\phi_2(P)$ denote its $2$-dimensional flow number.

Petersen graph flow-number conjecture.

$$\phi_2(P)=1+\sqrt{7/3}.$$

The paper gives the matching upper bound and proposes equality based on computational evidence; no proof of the lower bound is known.