Identities between q-hypergeometric and hypergeometric integrals of different dimensions

Given complex numbers $m_1,l_1$ and nonnegative integers $m_2,l_2$, such that $m_1+m_2=l_1+l_2$, for any $a,b=0,\mathrm{.}\mathrm{.}\mathrm{.},\min (m_2,l_2)$ we define an $l_2$-dimensional Barnes type q-hypergeometric integral $I_{a,b}(z,\mu ;m_1,m_2,l_1,l_2)$ and an $l_2$-dimensional hypergeometric integral $J_{a,b}(z,\mu ;m_1,m_2,l_1,l_2)$. The integrals depend on complex parameters $z$ and $\mu$. We show that $I_{a,b}(z,\mu ;m_1,m_2,l_1,l_2)$ equals $J_{a,b}(e^{\mu },z;l_1,l_2,m_1,m_2)$ up to an explicit factor, thus establishing an equality of $l_2$-dimensional q-hypergeometric and $m_2$-dimensional hypergeometric integrals. The identity is based on the $(g{l}_k,g{l}_n)$ duality for the qKZ and dynamical difference equations.