The negative immersions to local quasiconvexity conjecture

Let $X$ be a compact $2$-complex with negative immersions. Write $\pi_1X$ for its fundamental group, and let local quasiconvexity and the finite-generation intersection property relative to $\mathcal{H}$ be the properties denoted in the source by

$$and$$

. The negative immersions to local quasiconvexity conjecture. Negative immersions should imply local quasiconvexity, and hence should imply the finite-generation intersection property relative to $\mathcal{H}$. The source states this as an open conjectural implication; no resolution is given.