The coefficient characterization conjecture for divisor-sum congruences

Let $A$ and $B$ be nonnegative integers satisfying $0\leq B<A$, and let $N=An+B$. Define $\sigma_0(m)$ to be the number of positive divisors of $m$. Suppose that, for every nonnegative integer $n$,

$$\sum_{1\leq k<\sqrt{N}}\sigma_0(N-k^2)\equiv0\pmod4.$$

The coefficient characterization conjecture. Under this assumption, $A\equiv0\pmod4$ and $B\equiv2\pmod4$.

This conjecture is based on a computational observation about pairs $(A,B)$ for which the displayed congruence holds throughout the authors' search range. The phenomenon is not explained by the approach of the paper or by the previously mentioned work, so the asserted necessity of these congruence conditions remains open.