(Hypothesis) For integer $m \geq 1$, $l_j > 0$, and $x_j \in (-1,1)$, then the following identity, whether or not to be established:
$$ \prod_{j=1}^m \text{Li}_{l_j}\left[ x_j \right] = \sum_{k=0}^m \sum_{(i_1, i_2, \ldots, i_k) \in \{1,2, \ldots, m\}, i_1 < i_2 < \ldots < i_k} (-1)^{m-1-k} \cdot S\left[ \begin{array}{c} {l_{i_1}}, \ldots, {l_{i_k}} \\ {x_{i_1}}, \ldots, {x_{i_k}} \end{array} \middle| \begin{array}{c} {\sum_{j=1}^m {l_j} - \sum_{j=1}^k {l_{i_j}}} \\ {x_1 \cdots x_m / (x_{i_1} \cdots x_{i_k})} \end{array} \right] $$
with $S\left[ \begin{array}{c} \emptyset \\ \emptyset \end{array} \middle| \begin{array}{c} l \\ x \end{array} \right] := \text{Li}_l\left[ x \right]$, where $\text{Li}_p\left[ x \right]$ denotes the q-polylogarithm function defined by:
$$ \text{Li}_p\left[ x \right] := \sum_{n=1}^\infty \frac{{x^n}}{{(n)_q^p}} $$
and the sums $S\left[ \begin{array}{c} {l_1}, \ldots, {l_{m-1}} \\ {x_1}, \ldots, {x_{m-1}} \end{array} \middle| \begin{array}{c} {l_m} \\ {x_m} \end{array} \right]$ are defined by:
$$ S\left[ \begin{array}{c} {l_1}, \ldots, {l_{m-1}} \\ {x_1}, \ldots, {x_{m-1}} \end{array} \middle| \begin{array}{c} {l_m} \\ {x_m} \end{array} \right] := \sum_{n=1}^\infty \frac{{\zeta_n[l_1,x_1] \zeta_n[l_2,x_2] \cdots \zeta_n[l_{m-1},x_{m-1}]}}{{(n)_q^l}}x_m^n $$
Here, the finite sums $\zeta_n[l,x]$ are called the partial sum of q-polylogarithm function defined by:
$$ \zeta_n[l,x] = \sum_{j=1}^n \frac{{x^j}}{{(j)_q^l}} $$
and the non-negative integer $(n)_q$ is defined as:
$$ (n)_q := \frac{{1 - q^n}}{{1 - q}}, \quad 0 < q < 1. $$
I have proved that when $m \leq 5$, the hypothesis is true. However, how to prove the general solution?