Romik's conjecture on the p-adic properties of the sequence d(n)

Let $d(n)$ be the integer sequence defined by

$$(1-w)^{-1/2}\theta_3\left(\frac{i+wi}{1-w}\right)=\theta_3(i)\sum_{n=0}^{\infty}\frac{d(n)}{(2n)!}\left(\frac{\gamma(1/4)^4}{8\pi^2\sqrt{2}}w\right)^{2n},\qquad |w|<1,$$

where $\theta_3(z):=\sum_{n=-\infty}^{\infty}e^{\pi i n^2z}$. Romik's conjecture. Let $p$ be an odd prime. Then:

1. If $p\equiv3\pmod 4$, then $d(n)\equiv0\pmod p$ for sufficiently large $n$.
2. If $p\equiv1\pmod 4$, the sequence $\{d(n)\pmod p\}_{n=0}^{\infty}$ is periodic.
3. If $p\equiv3\pmod 4$ and $m\ge2$, then $d(n)\equiv0\pmod {p^m}$ for sufficiently large $n$.
4. If $p\equiv1\pmod 4$ and $m\ge2$, the sequence $\{d(n)\pmod {p^m}\}_{n=0}^{\infty}$ is periodic.

These assertions describe the expected eventual vanishing or periodicity of the Taylor coefficients modulo powers of odd primes, extending the known integrality of $d(n)$. The conjecture is attributed to Romik and is presented here as open.