Riesz bases of reproducing kernels in Fock type spaces

In a scale of Fock spaces $\mathcal F_\varphi$ with radial weights $\varphi$ we study the existence of Riesz bases of (normalized) reproducing kernels. We prove that these spaces possess such bases if and only if $\varphi(x)$ grows at most like $(\log x)^2$.