Logarithmic growth conjecture for all classes of consecutive powerful-number progressions

Let $\mathcal{A}$ be the set of positive integers $N$ for which the corresponding triple $N,N+d,N+2d$ is a three-term arithmetic progression of consecutive powerful numbers. Partition it as

$$\mathcal{A}=\mathcal{A}_0\sqcup\mathcal{A}_1\sqcup\mathcal{A}_2,$$

where $\mathcal{A}_i$ consists of those $N$ for which the triple contains exactly $i$ squares. Logarithmic-growth conjecture. For every $i\in\{0,1,2\}$,

$$\left|\mathcal{A}_i\cap\{1,2,\ldots,n\}\right|\asymp\log n.$$

In particular,

$$\left|\mathcal{A}\cap\{1,2,\ldots,n\}\right|\asymp\log n.$$

The conjecture is motivated by heuristic probability estimates for the three classes and remains unproved.