The normalish subgroup conjecture for Thompson's group $T$

Let $T$ be R. Thompson's group, and let a subgroup be normalish when every finite intersection of its distinct conjugates is infinite. An embedded copy of a group means a subgroup isomorphic to that group.

Normalish subgroup conjecture. Every normalish subgroup of $T$ contains an embedded copy of R. Thompson's group $F$ or an embedded non-abelian free subgroup.

The conjecture is motivated by restrictions on subgroups of $T$ that contain neither a copy of $F$ nor non-abelian free subgroups, together with Brin's Ubiquity Theorem and a lemma of Bleak, Kassabov, and Moore. The source gives no resolution of this question.