Second Borwein conjecture on coefficient positivity

For each positive integer $n$, define polynomials $\alpha_n(q)$, $\beta_n(q)$ and $\gamma_n(q)$ by

$$\prod_{j=1}^n(1-q^{3j-2})^2(1-q^{3j-1})^2=\alpha_n(q^3)-q\beta_n(q^3)-q^2\gamma_n(q^3).$$

Second Borwein conjecture. Each of $\alpha_n(q)$, $\beta_n(q)$ and $\gamma_n(q)$ has non-negative coefficients.

This is the second of the three Borwein conjectures discussed in the source, which studies asymptotic sums of related coefficients. The supplied source does not state whether this conjecture has been resolved.