Asymptotic conjecture for $(2,k,k)$-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 $n$.

Asymptotic conjecture for $(2,k,k)$-numbers. For each $k$,

$$\np_{(2,k,k)}(n)=\frac{n}{(\log n)^{1/2}}G_k+R(n),\qquad G_k=\sqrt{\frac{2\log2}{\pi(k^2+k)}},\qquad R(n)=o\left(\frac{n}{(\log n)^{1/2}}\right).$$

This is part of the source's heuristic predictions based on a random model for binary digit counts. The formula and error estimate are not proved in the source and remain open.