Positive density conjecture for the nonzero coefficients of $\bar{B}(n)$

Let $\bar{B}(n)$ be the sequence defined in the paper, and let $X$ tend to infinity. Positive density conjecture for $\bar{B}(n)$. There exists a constant $\sigma_2\in(0,1]$ such that

$$\#\{n\le X:\bar{B}(n)\neq0\}\sim \sigma_2 X.$$

Identities relating $\bar{B}(n)$ to a positive ternary quadratic form show that representation by such a form is a necessary condition for $\bar{B}(n)$ to be nonzero. Theorems 3.3 and 3.4 provide a stepping stone, but the conjectured asymptotic and the related upper arithmetic density remain open.