The leafless vertex metric dimension bound conjecture

Let $G$ be a graph distinct from the cycle $C_n$, with minimum degree $\delta(G)\geq2$. Here $\mathrm{dim}(G)$ denotes the vertex metric dimension and $c(G)$ the cyclomatic number. The leafless vertex metric dimension bound conjecture.

$$\mathrm{dim}(G)\leq2c(G)-1.$$

The bound is known for leafless cactus graphs, and the source proves it for $\Theta$-graphs; the general case reduces to 2-connected graphs distinct from cycles and remains open.