Convexity question for the bounding function of normalized asymptotic translation length

Let $M$, $\mathcal{F}$, $d$, and $\Omega$ be as in the closure theorem: $M$ is a connected cusped hyperbolic $3$-manifold, $\mathcal{F}$ is a fully-punctured fibered face, and $\Omega$ is a rational $d$-dimensional slice with $d\geq 1$. Let $g\colon \operatorname{int}(\Omega)\to\mathbb{R}_+$ be the continuous function whose graph describes the accumulation points of the graph of the normalized asymptotic translation length function $\mu_d$.

Convexity question. Is the function $g$ convex?

The function $g$ is the boundary-diverging function controlling the accumulation behavior of $\mu_d$; its convexity is not clear from the explicit formula. This is posed as an open question.