Displacement-angle characterization of the negative mapping class semigroup

Let $\operatorname{Map}_{g,1}$ be the mapping class group of a genus-$g$ surface with one marked point, and let $\operatorname{Map}_{g,1}(-)\subset\operatorname{Map}_{g,1}$ be the semigroup under consideration. The central extension associated with the boundary action gives a homomorphism

$$\rho^{\operatorname{per}}:\operatorname{Map}_{g,1}\to\operatorname{Homeo}^{\operatorname{per}}(\mathbb R),$$

where $\operatorname{Homeo}^{\operatorname{per}}(\mathbb R)$ is the group of $2\pi$-periodic orientation-preserving homeomorphisms of $\mathbb R$. For $\phi\in\operatorname{Map}_{g,1}$ and $\ell\in\mathbb R$, its displacement angle is $(\rho^{\operatorname{per}}(\phi)(\ell)-\ell)\bmod 2\pi\mathbb Z$. Displacement-angle characterization. The subsemigroup $\operatorname{Map}_{g,1}(-)\subset\operatorname{Map}_{g,1}$ consists of mapping classes whose displacement angle is non-negative at every point of $\mathbb R$. This proposed characterization relates the semigroup to the boundary action of the mapping class group and generalizes the preceding displacement-angle criterion; the source presents it as conjectural, with no resolution supplied here.