Convexity conjecture for the minimal energy of doubly connected domains

Let $\Omega^*$ be a bounded doubly connected domain in $\mathbb{C}$, and let $\mathsf{E}(\tau,\Omega^*)$ denote the infimum of the Dirichlet energy over deformations from a doubly connected domain of modulus parameter $\tau$ onto $\Omega^*$. Convexity conjecture. The function

$$\tau\mapsto \mathsf{E}(\tau,\Omega^*)$$

is convex for $0<\tau<\infty$. This is motivated by the convexity theorem and the circular-annulus example; the claim would follow from the affirmative answer to the preceding question about equality of the homeomorphic and deformation infima.