The master pentagon identity for Ding-Iohara-Miki algebra generators

Let $U_{q_1,q_2,q_3} (\widehat{\widehat{\mathfrak{gl}}}_1)$ be the Ding-Iohara-Miki algebra, with standard generators $e_{\vec{\gamma}}$ indexed by $\vec{\gamma} \in \mathbb{Z}^2 \backslash \{(0,0)\}$. For each such $\vec{\gamma}$, define the formal generating function

$$T_{\vec{\gamma}}(u)=\exp\left(-\sum_{n\geq 1}\frac{(-u)^n}{n}e_{n\vec{\gamma}}\right).$$

Master pentagon identity. The generating functions satisfy

$$T_{(0,1)}(v)T_{(1,0)}(u)=T_{(1,0)}(u)T_{(1,1)}(uv)T_{(0,1)}(v).$$

This identity is proposed as a general pentagon identity for group-like elements in the Ding-Iohara-Miki algebra, encompassing the quantum dilogarithm pentagon identity and the five-term relation for certain operators related to Macdonald polynomials; the paper reports checks but does not establish the identity in full.