Quantitative almost-everywhere density conjecture for infinite subsets under interval exchanges

Let $I=[0,1]$, let $X$ be an infinite subset of $I$, and let $T$ be an interval exchange transformation (IET) of $I$. Quantitative density conjecture. For almost every IET $T$,

$$\inf_n n\,d^H\left(\bigcup_{k=0}^{n-1}T^kX,I\right)=0.$$

This is a quantitative strengthening concerning the Hausdorff-density rate of finite orbit unions of infinite sets under interval exchanges. The supplied text gives no resolution or partial result for this claim.