Kim–Lim–Lovejoy odd-balanced unimodal sequence congruence conjecture

Let $v(n)$ count odd-balanced unimodal sequences of size $2n+2$, and let $v(m,n)$ count those with rank $m$. Let $\ell\not\equiv-1\pmod8$ be prime and let $k$ be a positive integer satisfying

$$\ell\mid\mid 8k+7.$$

If $(8k+7)/\ell$ is a quadratic residue modulo $\ell$, then Kim–Lim–Lovejoy's conjecture.

$$v(\ell^2n+k)\equiv0\pmod4.$$

The same congruences were conjectured for the coefficients of $\mathcal V(\pm i,q)$. The stated version is false: the paper gives $\ell=17$, $k=99$ as a counterexample, so primes congruent to $1$ modulo $8$ must be excluded. A corrected version is proved later in the paper.