The cycle rank conjecture for metric dimension and zero forcing number

Let $G$ be a graph. Write $\dim(G)$ for its metric dimension, $Z(G)$ for its zero forcing number, and $r(G)$ for its cycle rank. Cycle Rank Conjecture.

$$\dim(G)\leq Z(G)+r(G).$$

The conjecture proposes that the difference between metric dimension and zero forcing number is bounded by the cycle rank. The paper gives examples showing that the analogous bound using the even cycle rank is false, while this cycle-rank bound is presented as an open conjecture.