The equality characterization for leafless vertex metric dimension

Let $G$ be a connected graph with minimum degree $\delta(G)\geq2$. A daisy graph is a graph consisting of at least two cycles sharing one vertex, and a petal is one of its cycles; $\Theta_{p,q,r}$ denotes the corresponding theta graph with three internally disjoint paths of lengths $p,q,r$. The leafless vertex metric dimension equality conjecture.

$$\mathrm{dim}(G)=2c(G)-1$$

if and only if $G$ is a daisy graph without odd petals, $G=\Theta_{p,p,p}$, or $G=\Theta_{p,p,p+2}$. The source presents this as an expected characterization; it is proved only for the classes treated in the paper, so the general statement remains open.