Andrews–El Bachraoui's parity conjecture for the odd companion series

Let $t_o(n)$ be defined by

$$\sum_{n=0}^\infty t_o(n)q^n:=T_o(q)=\sum_{m=0}^\infty \frac{(-q;q)_{2m}}{(q;q^2)_{m+1}}q^{2m},$$

where $q$ is a complex number with $|q|<1$ and $(a;q)_n$ denotes the standard $q$-Pochhammer symbol. Andrews–El Bachraoui's conjecture. If $8n+9$ has a prime divisor $p\equiv 5,7\pmod 8$ raised to an odd exponent, equivalently, if $8n+9$ is not represented by $x^2+2y^2$, then

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

This conjecture concerns the parity of the coefficients of the odd companion series associated with the two-color partition generating function $S_1(q)$. The supplied source presents it as a conjecture of Andrews and El Bachraoui; its resolution is not established by the provided context.