Companion Rogers–Ramanujan type identity in rank four (Conjecture 1.4)

Prove the Rogers–Ramanujan type identity for the rank-four Nahm-type triple sum: ∑_{n1,n2,n3 ≥ 0} q^{4 n1^2 + 4 n1 n2 + 3 n2^2 − 2 n2 n3 + n3 + 4 n1 + 2 n2} / [(q^4; q^4)_{n1} (q^4; q^4)_{n2} (q^4; q^4)_{n3}] = [(q^6; q^6)_∞ (q^8; q^8)_∞ (q^2, q^{10}; q^{12})_∞] / [ (q^4; q^4)_∞^2 (q, q^{11}; q^{12})_∞ (q^5, q^7; q^{12})_∞ ], where (a; q)_n = ∏_{k=0}^{n-1} (1 − a q^k) and (a; q)_∞ = ∏_{k=0}^{∞} (1 − a q^k).

Background

To evaluate rank-four tadpole Nahm sums, the authors establish several new Rogers–Ramanujan type identities (Theorem 1.3). They also identify a closely related companion identity (Conjecture 1.4) that they were unable to prove.

A proof of Conjecture 1.4 would complete the set of identities associated to the duals of a specific rank-three example in Zagier’s list, strengthening the modularity results derived from Theorem 1.3.