Covering radius conjecture for graph critical sets

Let $G$ be an undirected, connected multigraph with genus $g\geq 1$ and $n\geq 2$ vertices. Let $\mathcal{N}_G$, ${\rm Crit}_{\triangle}(L_G)$ and ${\rm Crit}_{\bar{\triangle}}(L_G)$ be the three sets associated with the graph Laplacian, and let $P_{1,\lambda}$ be the stated polytope. Covering radius conjecture for graph critical sets. For every $\lambda\in[1/g,g]$, the covering radii of all three sets with respect to the distance function induced by $P_{1,\lambda}$ are at least

$$\frac{\sqrt{g/\lambda}}{n}.$$

The proposition preceding the conjecture identifies the relevant covering radii, while the conjecture extends the non-special-divisor formulation to the two critical sets.