Asymptotic formula for leading-digit-restricted squares

Let $B\in\mathbb{N}_{\ge 2}$ and let $D\subseteq\{0,1,\dots,B-1\}$ satisfy $|D|\ge 2$. Let $\alpha_B^D(n)$ be the number of integers $B^{n-\frac12}\le x<B^{n+\frac12}$ such that the first $n$ base-$B$ digits of $x^2$ lie in $D$. Leading-digit asymptotic conjecture. There exists a constant $c\in\mathbb{R}_{>0}$ such that

$$\lim_{n\to\infty}\frac{\alpha_B^D(n)}{|D|^n}=c.$$

The preceding bounds establish exponential-order growth but not convergence to a normalized positive constant; the source gives no resolution.