Linear finite difference operators preserving Laguerre-P\'olya class

We completely describe all finite difference operators of the form $$\Delta_{M_1, M_2, h}(f)(z)=M_1(z) f(z+h) + M_2(z) f(z-h)$$ preserving the Laguerre-P\'olya class of entire functions. Here $M_1$ and $M_2$ are some complex functions and $h$ is a nonzero complex number.