Linear growth conjecture for genetic pseudo-Anosov braid closures

Let $B_{2n}$ be the braid group on $2n$ strands, let $\beta\in B_{2n}$ with $n\geq 3$ be a genetic pseudo-Anosov braid, and let $M$ and $k$ be the parameters defining the sequence of braid powers. Write $\widehat{\beta^{mk+1}}$ for the braid closure and $S$ for the associated Heegaard surface. The entropy of a pseudo-Anosov braid is the logarithm of its stretch factor.

Linear growth conjecture. The entropy of braids in the sequence $\\{\beta^{mk+1}\\}_{m=M+1}^\infty$, a lower bound on the knot genus of $\\{\widehat{\beta^{mk+1}}\\}_{m=M+1}^\infty$, the Hempel distances of $\\{(\widehat{\beta^{mk+1}},S)\\}_{m=M+1}^\infty$, and the hyperbolic volumes of $\\{\widehat{\beta^{mk+1}}\\}_{m=M+1}^\infty$ are all linearly related by a function in $m$.

The conjecture proposes a common linear dependence on the power parameter for dynamical complexity, knot genus, Heegaard-splitting distance, and hyperbolic volume. The surrounding discussion presents these as open questions; no resolution is given here.