The finite Dehn twist representation cohomology conjecture

Let $S_g^n$ be a compact oriented surface of genus $g$ with $n$ boundary components, and let $\mathrm{Mod}(S_g^n)$ denote its mapping class group. A representation is a finite Dehn twist representation if every Dehn twist has finite order under the representation; it is semisimple if the corresponding representation is semisimple. Finite Dehn twist representation conjecture. For any $g\geq 3$, $n\geq 0$, and semisimple finite Dehn twist representation $\rho$ of $\mathrm{Mod}(S_g^n)$,

$$H^1(\mathrm{Mod}(S_{g,n}),\rho)=0.$$

The conjecture is motivated by Litt's rigidity conjecture and by restrictions on representations of mapping class groups. In genus at least $3$, finite Dehn twist representations are equivalent to representations sending Dehn twists to diagonalizable elements, and unitary representations provide important examples; the conjecture remains open in the supplied text.