The explicit subaction conjecture associated with the period-two maximizing orbit

Let $A$ and $T(x)=2x\pmod 1$ be as in the preceding construction, with inverse branches $\tau_1(x)=x/2$ and $\tau_2(x)=(x+1)/2$. Define $\eta=\tau_2\circ\tau_1$ and

$$H(x)=\sum_{i=1}^{+\infty}\left(A\left(\tau_1(\eta^i(x))\right)+A\left(\tau_2(\tau_1(\eta^i(x)))\right)\right).$$

A subaction is a function satisfying the corresponding ergodic-optimization inequality. The explicit period-two subaction conjecture. The function

$$W(x)=H(x)I_{[0,1/2)}(x)+H(1-x)I_{[1/2,1]}(x)$$

is a subaction for $A$. This candidate is associated with the maximizing probability supported on $\{1/3,2/3\}$. The source says that the relevant series is absolutely convergent and seeks to prove the subaction identity, but supplies no proof of the conjecture.