Subdivision invariance conjecture for graph Brill–Noether numbers

Let $G$ be a graph, and let $\Gamma$ be the associated $\mathbb Q$-graph. For an integer $r\geq 1$, let $D(G,r)$ and $D(\Gamma,r)$ be the minimal degrees of a $g^r_d$ on $G$ and $\Gamma$, respectively. Let $\sigma_k(G)$ denote the $k$-fold subdivision of $G$. Subdivision invariance conjecture. For every $r\geq 1$,

$$D(G,r)=D(\sigma_k(G),r)\quad\text{for all }k\geq 1,$$

and

$$D(G,r)=D(\Gamma,r).$$

The conjecture is supported by computations, including verification of the first assertion for numerous graphs and small values of $r$ and $k$. Together with the proved Brill–Noether theorem for metric $\mathbb Q$-graphs, it would imply the nonnegative-Brill–Noether-number part of the graph conjecture.