The faithfulness conjecture for the matrix representation of Clifford groups

Let $\mathcal{C}_n$ be the $n$-qubit Clifford group, and let $\mathcal{M}_{2^n}$ denote the matrix representation obtained from its action on the $2^n$-dimensional qubit space.

Faithfulness conjecture. The matrix representation $\mathcal{M}_{2^n}$ for $\mathcal{C}_n$ is faithful.

Faithfulness is useful in representation theory: for a faithful $G$-module $V$, every irreducible $G$-module occurs in at least one tensor power $V^{\otimes r}$ for $0\leq r\leq m-1$, where $m$ is the number of distinct character values of $V$.