Andrews–Garvan–Straub conjecture on linear congruences for the function $\xi(n)$

Let $p$ be a prime, let $\xi(n)$ denote the function studied in the paper, and let $A,n\in\mathbb N$.

Andrews–Garvan–Straub conjecture. There exists a $B\in\mathbb N$ such that

$$\xi\left(p^A n-B\right)\equiv0\pmod{p^A}$$

for all $n$ precisely when

$$p=23\quad\text{or}\quad p\equiv 5,7,10,11,14,15,17,19,20,21,22\pmod {23}.$$

This conjecture asserts that the congruences already exhibited for $\xi(n)$ account for all linear congruences of the indicated form. The source describes it as an important challenge for future work, and gives no resolution.