A factorial-ratio q-congruence for powers of an odd integer

Let $n\equiv1\;\allowbreak(\operatorname{mod}4)$ be an integer greater than $1$, and let $r,s$ be positive integers with $r>s$. For $(a;q)_k$ the q-shifted factorial, the factorial-ratio conjecture asserts

$$\frac{(q;q^2)_{(n^r-1)/4}(q^{2n};q^{2n})_{(n^{r-1}-1)/4}}{(q^2;q^2)_{(n^r-1)/4}(q^n;q^{2n})_{(n^{r-1}-1)/4}} \equiv \frac{(q;q^2)_{(n^s-1)/4}(q^{2n};q^{2n})_{(n^{s-1}-1)/4}}{(q^2;q^2)_{(n^s-1)/4}(q^n;q^{2n})_{(n^{s-1}-1)/4}} \;\allowbreak(\operatorname{mod}\Phi_{n^s}(q)^2).$$

The source presents this as a conjectural q-congruence that would imply the strengthening of Lemma 3; no resolution is given.