The Second Borwein sign-pattern conjecture for squares

Let $n$ be a positive integer, and let

$$P_n(q):=(1-q)(1-q^2)(1-q^4)(1-q^5)\cdots(1-q^{3n-2})(1-q^{3n-1}).$$

The Second Borwein conjecture. The sign pattern of the coefficients in the expansion of $P_n(q)^2$ is $+--+--+--\cdots$, with the same convention that a zero coefficient is considered as both $+$ and $-$. The source states that this conjecture has been proved.