Keith–Zanello's self-similarity conjecture for 19-regular partitions

Let $b_{19}(n)$ denote the number of partitions of $n$ with no parts divisible by $19$. For a prime $p>3$, let $b3$ be the integer satisfying

$$\gamma\equiv-3\cdot8^{-1}\pmod p,\qquad0<\gamma<p,$$

and set

$$\delta=\left\lfloor\frac{3p}{8}\right\rfloor.$$

Keith–Zanello's self-similarity conjecture. For a positive proportion of primes $p$, one has

$$\sum_{n=0}^{\infty}b_{19}(2(pn+\gamma))q^n\equiv q^\delta\sum_{n=0}^{\infty}b_{19}(2n)q^{pn}\pmod2.$$

This conjecture predicts a self-similarity in the parity of 19-regular partition numbers for a positive proportion of primes; the supplied source does not state whether it has been resolved.