Poor-edge strengthening of the Petersen Coloring Conjecture

Let $G$ be a bridgeless cubic graph, and let $\mathrm{NC}(G)$ be the set of all normal $5$-colorings of $G$. Assuming the Petersen Coloring Conjecture, define $\mathrm{poor}(G)$ as the maximum number of poor edges among colorings in $\mathrm{NC}(G)$. Let $P_{10}$ be the Petersen graph, and let $P_{10}^{\Delta}$ be obtained from $P_{10}$ by truncating one vertex. Poor-edge strengthening of the Petersen Coloring Conjecture. If $G\ne P_{10}$, then $\mathrm{poor}(G)>0$. Moreover, if $G\ne P_{10},P_{10}^{\Delta}$, then $\mathrm{poor}(G)\geq 6$.

This proposed strengthening concerns the abundance of poor edges in normal $5$-colorings beyond the Petersen graph and its one-vertex truncation. The source attributes it to earlier work, but gives no resolution; its status is therefore open.