Cycle-rank conjecture for vertex metric dimension

Let $G$ be a connected graph. Let $L(G)$ denote the number of leaves, and let $c(G)=\left\vert E(G)\right\vert-\left\vert V(G)\right\vert+1$ be its cyclomatic number. The cycle-rank conjecture for vertex metric dimension.

$$\mathrm{dim}(G)\leq L(G)+2c(G).$$

The paper proves this bound for cactus graphs and notes that it also holds for $3$-connected graphs; its validity for all connected graphs is left as a conjecture.