Boshernitzan–Kornfeld conjecture on the prevalence of finite type interval translation maps

Let $r\geqslant 2$, and let $\mathrm{ITM}(r)$ denote the space of interval translation maps on $r$ intervals. An interval translation map is of infinite type when, for its attractor construction $X_n=T^n(I)$, one has $X_{n+1}\subsetneq X_n$ for every $n$. Boshernitzan–Kornfeld conjecture. For all $r\geqslant 2$, the set of all infinite type $\mathrm{ITM}$s on $r$ intervals is a measure zero subset of $\mathrm{ITM}(r)$. The conjecture asserts that infinite type behavior is rare among interval translation maps; equivalently, finite type should be prevalent in measure for every number of intervals. The source gives no resolution, so the conjecture remains open.