Parity conjecture for the unique invariant theta characteristic on modular curves

Let $p$ be an odd prime with $p\geq 7$, and let $X(p)$ denote the modular curve of level $p$. An invariant theta characteristic is a theta characteristic on $X(p)$ invariant under the relevant group action.

Parity conjecture. The unique invariant theta characteristic on $X(p)$ is always even.

This conjecture extends the computations for $p=7,11,13,17$, where the unique invariant theta characteristic was found to be even. The parity for all odd primes $p\geq 7$ remains open.