Hirschhorn–Sellers conjecture on the parity of $2^t$-core partition numbers

Let $t\geq 2$. For each $n\geq 1$ with $3\nmid n$, let $a_{2^t}(n)$ denote the number of $2^t$-core partitions of $n$, whose generating function is

$$\sum_{n\geq 0}a_{2^t}(n)q^n=\prod_{n\geq 1}\frac{(1-q^{n2^t})^{2^t}}{1-q^n}.$$

Hirschhorn–Sellers conjecture. For all $n\geq 1$ with $3\nmid n$,

$$a_{2^t}\left(\frac{3^{2^{t-1}-1}n-\frac{4^t-1}{3}}{8}\right)\equiv 0\pmod{2}.$$

This conjecture, proposed by Hirschhorn and Sellers in 1999, asserts an evenness property for the number of $2^t$-core partitions at a specified arithmetic progression of arguments.