Keith's parity conjecture for reciprocals of false theta functions

Let $\Psi(a,b)$ be the false theta function

$$\Psi(a,b):=\sum_{n=0}^\infty a^{n(n+1)/2}b^{n(n-1)/2}-\sum_{n=-\infty}^{-1}a^{n(n+1)/2}b^{n(n-1)/2}.$$

For integers $r,s$, define $c_{r,s}(n)$ by

$$\sum_{n=0}^\infty c_{r,s}(n)q^n=\frac{1}{\Psi(-q^r,q^s)}.$$

Keith's conjecture. For $n\geq 0$,

$$c_{13,1}(32n+23)\equiv 0 \pmod 2,$$ $$c_{13,1}(64n+63)\equiv 0 \pmod 2,$$

and

$$c_{13,1}(72n+j)\equiv 0 \pmod 2,$$

where $j\in\{15,21,39,69\}$. This conjecture predicts infinite families of parity congruences for coefficients of reciprocals of false theta functions; the supplied source gives no resolution, so its status remains open.