A family of q-congruences for truncated ${}_3\phi_2$ series when n is 2 modulo 3

Let $(a;q)_k=(1-a)(1-aq)\cdots(1-aq^{k-1})$ be the $q$-shifted factorial, let $\Phi_n(q)$ denote the $n$-th cyclotomic polynomial, and for a rational number $x$ whose denominator is coprime with $n$, let $\langle x\rangle_n$ denote the least non-negative residue of $x$ modulo $n$. Let $n$ be a positive integer with $n\equiv 2\pmod 3$, let $m$ be a positive integer with $\gcd(m,n)=1$, and let $r$ be an integer satisfying

$$0<\left\langle\frac{r}{3m}\right\rangle_n\leqslant\frac{2n-1}{3}.$$

The conjecture. Then

$$\sum_{k=0}^{n-1}\frac{(q^m;q^{3m})_k(q^r;q^{3m})_k(q^{2m-r};q^{3m})_kq^{3mk}}{(q^{3m};q^{3m})_k^3}\equiv 0\pmod{\Phi_n(q)^2}.$$

The source presents this as one of two proposed general families of q-congruences and does not state that it has been resolved.