A q-congruence for a second truncated ${}_3\phi_2$ series modulo a cyclotomic square

Let $(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ be the $q$-shifted factorial, and let $\Phi_n(q)$ denote the $n$-th cyclotomic polynomial. For a positive integer $n$ with $n\equiv 5\pmod 9$, consider

$$\sum_{k=0}^{n-1}\frac{(q^{-1},q^{-2},q^{-6};q^9)_k q^{9k}}{(q^9;q^9)_k^3}.$$

The conjecture. The truncated series should satisfy

$$\sum_{k=0}^{n-1}\frac{(q^{-1},q^{-2},q^{-6};q^9)_k q^{9k}}{(q^9;q^9)_k^3}\equiv 0\pmod{\Phi_n(q)^2}.$$

This is one of the two nonsymmetric conjectures proposed in the concluding section for further study; no resolution is supplied in the source.