Convexity conjecture for the pendulum time map

Let $T_B(z,\phi)$ be the time map defined in the paper for parameters $z$ and $\phi$, with $z\in(0,2)$ and $\phi\in(0,\frac{\pi}{2})$. Convexity conjecture. For each $\phi\in(0,\frac{\pi}{2})$, the function $z\mapsto T_B(z,\phi)$ is convex. The authors report convincing numerical evidence for the positivity of the quantity governing the second derivative, but were unable to prove it; establishing this convexity would yield the existence of a single minimum of the time-map graph.