Optimality of asymptotic Nielsen number bounds for trellis maps

Let $T$ be a trellis and let $f$ be a trellis map for $T$. Write $N_\infty(f)$ for the asymptotic Nielsen number and $h_\mathit{top}$ for topological entropy. Optimality conjecture. $N_\infty(f)$ is a lower bound for all maps with trellis $T$ homotopic to $f$. Moreover, there is a homeomorphism homotopic to $f$ with topological entropy $N_\infty(f)$, and for every $\epsilon>0$ there is a uniformly hyperbolic diffeomorphism homotopic to $f$ such that

$$h_\mathit{top} h<N_\infty(f)+\epsilon.$$

The conjecture concerns the optimality of entropy bounds obtained from trellis methods. The surrounding discussion notes that such optimality holds for many examples but cannot hold in complete generality for arbitrary trellises, so the asserted statement remains open.