Guzman's group-theoretic conjecture for subgroups of free groups

Let $F$ be a free group, and let $H,K\le F$ be subgroups with equal rank $m\ge 2$. If

$$\operatorname{rk}(H\cap K)\ge m,$$

then Guzman's group-theoretic conjecture.

$$\operatorname{rk}(H\vee K)\le m.$$

Here $H\vee K$ is the subgroup generated by $H$ and $K$. The conjecture was introduced in connection with hyperbolic $3$-manifolds and is presented in the source as a conjectural implication for the relationship between intersection and join ranks.