The explicit subaction conjecture for the Cantor-distance potential

Let $K$ be the set used in the source, let $d(x,K)$ denote the distance from $x$ to $K$, and set $A(x)=-d(x,K)$. Let $T(x)=2x\pmod 1$ on $[0,1]$, with inverse branch $\tau_1(x)=x/2$. Define

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

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

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

is a subaction. The formula gives an explicit candidate subaction for the distance-to-$K$ potential under the doubling map. The source reports numerical evidence for calibrated-subaction behavior but does not establish the conjecture.