Rank-four tadpole Rogers–Ramanujan companion identity

Let $q$ be a formal variable, and for $n\geq0$ let $(q^4;q^4)_n$ and $(q^m;q^m)_\infty$ denote the usual finite and infinite $q$-Pochhammer products; products with several arguments are understood multiplicatively. Rank-four tadpole companion identity. The following identity is conjectured:

$$\sum_{n_1,n_2,n_3\geq0}\frac{q^{4n_1^2+4n_1n_2+3n_2^2-2n_2n_3+n_3^2+4n_1+2n_2}}{(q^4;q^4)_{n_1}(q^4;q^4)_{n_2}(q^4;q^4)_{n_3}}=\frac{(q^6;q^6)_{\infty}(q^8;q^8)_{\infty}(q^2,q^{10};q^{12})_{\infty}}{(q^4;q^4)_{\infty}^2(q,q^{11};q^{12})_{\infty}(q^5,q^7;q^{12})_{\infty}}.$$

The identity is intended as a companion to the proved rank-four Rogers–Ramanujan-type identities in the paper, but the authors state that they cannot prove it at present and leave it as an open problem.