Benson's quasi-polynomial conjecture for tensor powers

Let $V$ be an odd-dimensional indecomposable representation of a finite $2$-group $G$ over an algebraically closed field $\Bbbk$ of characteristic $2$. Assuming the relevant tensor powers have a unique odd-dimensional indecomposable summand, denote that summand in $V^{\otimes n}$ by $V_n$ and define $P_V(n)=\dim(V_n)$. Benson's tensor-powers conjecture. The function $P_V$ is quasi-polynomial: there exist polynomials $f_0,f_1,\dots,f_{m-1}$ such that $P_V(n)=f_i(n)$ whenever $n\equiv i\pmod m$. This conjecture is proved in the paper for several monomial representations, while it remains open in general.