Almost-everywhere density conjecture for infinite subsets under interval exchange transformations

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

$$\inf_n d^H(T^nX,I)=0.$$

This conjecture asks whether every infinite subset of the interval comes arbitrarily close to the whole interval under some iterate of almost every interval exchange transformation. The supplied text gives no resolution or partial result for this claim.