Bryson–Ono–Pitman–Rhoades mod 4 conjectures for strongly unimodal sequences

Let $u(n)$ count strongly unimodal sequences of size $n$, let $u(m,n)$ count those of rank $m$, and let $u(a,b;n)$ count those whose rank is congruent to $a$ modulo $b$. For a prime $\ell$ with $\ell$ congruent to $7,11,13,17\pmod{24}$ and satisfying the quadratic-character condition $\left(\frac{k}{\ell}\right)=-1$, set

$$s(\ell)=\frac{\ell^2-1}{24}.$$

Bryson–Ono–Pitman–Rhoades conjecture. For all $n$,

$$u(\ell^2n+k\ell-s(\ell))\equiv0\pmod4.$$

Moreover, for every $a\in\{0,1,2,3\}$,

$$u(a,4;\ell^2n+k\ell-s(\ell))\equiv0\pmod2,$$

and

$$u(0,4;\ell^2n+k\ell-s(\ell))\equiv u(2,4;\ell^2n+k\ell-s(\ell))\pmod4.$$

These congruences concern the rank distribution of strongly unimodal sequences and were conjectured as mod $4$ refinements of known parity results. The paper proves these conjectures.