Asymptotic for the Pell-family consecutive powerful-number progressions

Let $\mathcal{A}$ be the set of positive integers $N$ for which some $d\in\mathbb{N}$ makes $N,N+d,N+2d$ a three-term arithmetic progression of consecutive powerful numbers. Let $\mathcal{A}_2$ consist of those $N$ for which the corresponding progression contains exactly two squares. For the Pell-family progressions, write

$$N=(x-2)^2,\qquad N+d=(x-1)^2,\qquad N+2d=7^3y^2=x^2-2.$$

Here $A=130576328$. Pell-family asymptotic conjecture. The number of such progressions with all terms in $[1,n]$ is

$$\bigl(C_7+o(1)\bigr)\log n,$$

where

$$C_7=\frac{1}{2}\prod_{\substack{m\notin\{1,7\}\m\ \operatorname{squarefree}}}\left(1-\frac{2}{m^{3/2}}\right)\Bigl(\log\bigl(A+\sqrt{A^2-1}\bigr)\Bigr)^{-1}\approx0.0014.$$

This is a heuristic conjecture based on the recurrence and local-density considerations; no proof of the stated asymptotic is given.