The Andrews spt-function mod 4 conjecture

Let $\operatorname{spt}(n)$ denote the number of occurrences of the smallest part among the partitions of $n$. Let $\ell>3$ be prime with $\ell\not\equiv23\pmod{24}$, and define

$$\widetilde\varepsilon(\ell)= \begin{cases}1,&\ell\equiv1\pmod{24},\\-1,&\text{otherwise.}\end{cases}$$

Set $s(\ell)=(\ell^2-1)/24$. The Andrews spt-function mod 4 conjecture. If $\left(\frac{n}{\ell}\right)=\widetilde\varepsilon(\ell)$, then

$$\operatorname{spt}(\ell n-s(\ell))\equiv0\pmod4.$$

The conjecture was presented by the second author in a 2017 talk. The paper proves it, and relates it to the corresponding congruences for a third-order mock theta function.