Parity conjecture for coefficients of the odd triangular partition series

Let $T_o(q)=\sum_{n\geq 0}t_o(n)q^n$, and let $n$ be a nonnegative integer. Parity conjecture. If $8n+9$ has a prime divisor $p\equiv 5,7\pmod{8}$ to odd exponent, equivalently if $8n+9$ is not represented by $x^2+2y^2$, then

$$t_o(n)\equiv 0\pmod{2}.$$

This predicts evenness of the relevant coefficients at indices characterized by the arithmetic condition on $8n+9$. The supplied text does not state whether the conjecture has been proved or disproved.