Logarithmic-density conjecture for binary partition values that are sums of two squares

Let $\mathcal{T}(x)$ count the integers $n\leq x$ such that $b(n)$ is a sum of two squares, where $b(n)$ is the binary partition function. Logarithmic-density conjecture. There exists a positive real number $c$ such that

$$\mathcal{T}(x)=c\frac{x}{\log x}+O\left(\frac{x}{\log^2 x}\right)$$

as $x\rightarrow+\infty$. This heuristic is based on the growth estimate $\log_2 b(n)\approx \frac12(\log_2 n)^2$ and the expected probability that $b(n)$ is a sum of two squares; the asymptotic formula remains unproved.