Merca's conjecture on divisibility of divisor sums in arithmetic progressions

For an integer $k\geq0$, let

$$\sigma_0(n)=\sum_{d\mid n}1$$

denote the number of positive divisors of $n$. Merca's conjecture. If

$$\sigma_0(An+B)\equiv0\,\,({\rm mod}\,\,{2^k})$$

for every $n\geq0$, then there is a sequence of prime numbers $p_1\leq p_2\leq\cdots\leq p_{k-1}$ such that $(p_1p_2\cdots p_{k-1})^2$ divides $A$ and $p_1p_2\cdots p_{k-1}$ divides $B$. This question is described in the source as too broad to be meaningfully resolved explicitly in this form.