The normal-subgroup conjecture for n-qubit Clifford groups

Let $n\geq 3$, let $\mathcal{C}_n$ be the $n$-qubit Clifford group, and let $\mathcal{P}_n$ be the $n$-qubit Pauli group. A normal subgroup of $\mathcal{C}_n$ is a subgroup invariant under conjugation by every element of $\mathcal{C}_n$.

Normal-subgroup conjecture. When $n\geq 3$, the $n$-qubit Pauli group $\mathcal{P}_n$ is the only non-trivial proper normal subgroup of $\mathcal{C}_n$.

This conjecture describes the expected normal-subgroup structure of the Clifford group in the indicated range of $n$; the supplied source material gives no resolution beyond presenting it as an observation expected to hold generally.