A cubic q-supercongruence with shifted factors

Let $m$ and $n$ be positive integers with $n$ odd, and let $(a;q)_k$ denote the q-shifted factorial. The cubic q-supercongruence conjecture.

$$\sum_{k=0}^{mn-1}(-1)^k[4k+1]\frac{(q^2;q^4)_k^3}{(q^4;q^4)_k^3}q^k \equiv \frac{[n]_{q^2}(-q^3;q^4)_{(n-1)/2}}{(-q^5;q^4)_{(n-1)/2}}(-q)^{(1-n)/2}\sum_{k=0}^{m-1}(-1)^k(4k+1)\frac{(\frac12)_k^3}{k!^3}\;\allowbreak(\operatorname{mod}\Phi_n(q)^3).$$

The $m=1$ case was proved by the author and Zudilin; the full statement is presented as a consequence for an earlier conjecture.