Refinement of the modular-degree parity theorem for Neumann–Setzer curves

Let $p=u^2+64$ be prime, and let $E_0$ be the Neumann–Setzer elliptic curve of conductor $p$. The modular degree of $E_0$ is the degree of its optimal modular parametrization from $X_0(p)$. The modular-degree refinement conjecture. If $u\equiv 7\pmod{8}$, then $2$ exactly divides the modular degree of $E_0$ if and only if $u\equiv 7\pmod{16}$. This is supported by experimental data, but the paper states that it is unclear whether its proof method extends to this refinement.