Typical weak-mixing conjecture for Bruin–Troubetzkoy interval translation maps

Let $d49c$ be the measure obtained in the theorem on unique ergodicity, and consider Bruin–Troubetzkoy interval translation maps of infinite type. Weak-mixing conjecture. Almost all (with respect to the measure $d49c$ obtained in the theorem on unique ergodicity) Bruin-Troubetzkoy ITMs of infinite type are weakly mixing. Weak mixing is known for many self-similar Bruin–Troubetzkoy ITMs of infinite type, and the authors state that they expect it to be typical; the supplied text gives no resolution of this conjecture.