The Third Borwein conjecture at modulus five

For a positive integer $n$, define the finite $q$-Pochhammer symbol $(q;q)_r:=\prod_{j=1}^{r}(1-q^j)$ and

$$S_n(q):=\frac{(q;q)_{5n}}{(q^5;q^5)_n}.$$

Third Borwein conjecture. The sign pattern of the coefficients in the expansion of $S_n(q)$ is $+----+----+----\cdots$, with the convention that zero coefficients may be regarded as either sign. The source discusses an approach to this conjecture but does not establish the full assertion.