The cobracket criterion for generalized Dehn twists

Let $S$ be a surface, let $C$ be an unoriented loop on $S$, and let $L(C)$ denote its associated invariant. Write $\delta$ for the cobracket on the completed loop space.

Cobracket criterion conjecture. If

$$\delta L(C)=0,$$

then $C$ is homotopic to a power of a simple closed curve.

This conjecture would answer the question of when a generalized Dehn twist can be realized by a diffeomorphism: Proposition 6.1 gives the vanishing of $\delta L(C)$ as a necessary condition, while the conjecture asserts that this condition forces the loop to come from a power of a simple closed curve.