Triple LLT product tableau expansion conjecture

Let $\bm{\nu}=(\lambda^1/\mu^1,\lambda^2/\mu^2,\lambda^3/\mu^3)$ be a triple of skew diagrams. For every triple $\{i,j,k\}=\{1,2,3\}$ with $i<j$, let $\operatorname{LLT}(\lambda^i/\mu^i,\lambda^j/\mu^j)(q)$ and $\operatorname{LLT}(\lambda^k/\mu^k)(q)$ be the corresponding LLT polynomials. Triple LLT product conjecture. One has

$$\operatorname{LLT}(\lambda^i/\mu^i,\lambda^j/\mu^j)(q)\cdot \operatorname{LLT}(\lambda^k/\mu^k)(q)=\sum_\lambda\left(\sum_{\substack{T\in\operatorname{RSST}(\lambda)\operatorname{Des}_3'(T)=D'(\beta)\c(\bm{\nu})\text{ - entries of }T}}q^{e_{i,j}(T)}\right)s_\lambda.$$

This is introduced as a belief rather than a formally named conjecture. The provided context does not define all notation in the asserted expansion, in particular $\beta$ and the precise meaning of the entry condition, so the claim and its status should be checked against the paper.