Juhász's depth bound conjecture for taut foliations

Let $(M,\gamma)$ be a taut balanced sutured manifold with $H_2(M)=0$. The sutured Floer homology $SFH(M,\gamma)$ is a finite-dimensional vector space over $\mathbb{Z}_2$, and let $k$ be such that

$$\operatorname{rk}_{\mathbb{Z}_2}(SFH(M,\gamma))<2^{k+1}.$$

Juhász's conjecture. The sutured manifold $(M,\gamma)$ admits a taut foliation of depth at most $2k$.

This conjecture seeks a bound on the depth of the taut foliations whose existence was established by Gabai. It relates the complexity of sutured Floer homology to the minimum possible depth, and its status is not resolved in the supplied source.