The finite-dimensional property (T) conjecture for mapping class groups

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. For a finite-dimensional unitary representation $\rho$ of this group, let $H^1(\mathrm{Mod}(S_g^n),\rho)$ denote group cohomology with coefficients in the associated representation. Finite-dimensional property (T) conjecture. For $g\geq 3$ and $n\geq 0$, every finite-dimensional unitary representation $\rho$ satisfies

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

This intermediate conjecture is intended to cover unitary quantum representations, including the $\mathrm{SU}(2)$ and $\mathrm{SO}(3)$ cases. It is related to Daniel Litt's conjecture that irreducible representations of mapping class groups in genus at least $3$ should be rigid, but no resolution is given here.