= Solution
Write $q_{ij}=c m_{ij}$ for $i\ne j$, with a common positive constant $c$, and put $d_i=\sum_{j\ne i}m_{ij}$. The <infinitesimal generator matrix> is symmetric because the graph is undirected. Thus the <uniform distribution> $\pi_i=1/|I|$ satisfies <detailed balance>, and the <continuous-time random walk> is reversible when started uniformly.
Its <jump chain> chooses a link uniformly from those incident to its current vertex:
$$
\widehat P_{ij}=\frac{m_{ij}}{d_i},\qquad \boxed{\widehat\pi_i=\frac{d_i}{\sum_jd_j}.}
$$
The flux $\widehat\pi_i\widehat P_{ij}=m_{ij}/\sum_jd_j$ is symmetric, so this <random walk on a multigraph> is also reversible. The continuous-time equilibrium is uniform because higher-degree vertices have shorter holding times; at jump times the equilibrium is degree-weighted. If the intended proportionality constant depends on the starting vertex, $q_{ij}=c_i m_{ij}$, reversibility still holds with $\pi_i\propto1/c_i$, and the same jump chain results.
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