The fixed-point conjecture for first-step primitive elements in PL homeomorphism groups

Let $I$ be the interval under consideration, and let $\Gamma$ be a finitely generated $C_0$-dense subgroup of $\operatorname{PL}_+(I)$. A first-step primitive element with respect to a generating set $S$ is a primitive element occurring after at most one Schreier transformation from $S$.

Fixed-point conjecture. There exists a generating set $S$ of $\Gamma$ such that every first-step primitive element with respect to $S$ fixes a point in $I$, while there exists $g\in\Gamma$ such that

$$\operatorname{Fix}(g)=\emptyset.$$

This conjecture extends the construction described for a particular pair of piecewise-linear homeomorphisms to every finitely generated $C_0$-dense subgroup of $\operatorname{PL}_+(I)$. The source does not state whether it has been resolved.