On a Multilinear Functional Equation

The following functional equation is solved:
$$f(x_1+z)\dotsb f(x_2+z)f(x_1+\dotsb+x_{s-1}-z) =\phi_1(x)\psi_1(z)+\dotsb+\phi_m(x)\psi_m(z),$$
where $x=(x_1,\dots,x_{s-1})$, for the unknowns $f,\psi_j\colon\mathbb C\to\mathbb C$ and $\phi_j\colon\mathbb C^{s-1}\to\mathbb C$ for $s\ge 3$ and $m\le 4s-5$.