The multiset-dimension upper-bound conjecture for graphs

Let $G$ be a graph on $n$ vertices. The multiset dimension $md(G)$ is the minimum cardinality of an m-resolving set when one exists, and $md(G)=\infty$ otherwise.

Multiset-dimension upper-bound conjecture. If $G$ has finite multiset dimension, then

$$md(G)\leq n-1.$$

This conjecture was proposed in the cited earlier work as a sharpening of the immediate bound $md(G)\leq n$. The paper proves the stronger bound $md(T)\leq n-2$ for trees of diameter at least $2$ with finite multiset dimension, partially settling the conjecture in the tree case; the general graph statement remains open.