The modulus-seven Borwein sign-pattern conjecture

For a positive integer $n$, define

$$\frac{(q;q)_{7n}}{(q^7;q^7)_n}=\sum_{m=0}^{21n^2}d_m(n)q^m.$$

Modulus-seven Borwein conjecture. For all $m$ and $n$, $d_{7m}(n)\ge0$ and $d_{7m+1}(n),d_{7m+3}(n),d_{7m+4}(n),d_{7m+6}(n)\le0$. Further, $d_{7m+5}(n)\ge0$ when $m\le3\alpha(n)n^2$ and $d_{7m+5}(n)\le0$ when $m>3\alpha(n)n^2$, where $\alpha(n)$ appears to stabilize around $0.302$.