Heuristic exponent conjecture for $(2,1,2)$-numbers

Let $B(n)$ denote the number of ones in the binary expansion of $n$, and let $p_{(k,l,m)}(n)$ count the $(k,l,m)$-numbers not exceeding $n$, where the relevant case consists of positive integers $n$ for which $B(n)=B(n^2)$.

Heuristic exponent conjecture for $(2,1,2)$-numbers.

$$\np_{(2,1,2)}(n)=n^{\alpha+o(1)},\qquad \alpha=\frac{\log 1.6875}{\log 2}\simeq0.7548875.$$

The prediction comes from treating $B(n)$ and $B(n^2)$ as independent random variables with equally likely zero and one outcomes. Known bounds in the source are weaker, so this asymptotic remains open.