Differential operators and entire functions with simple real zeros

Let ϕ and f be functions in the Laguerre-Pólya class. Write ϕ(z)=e−αz2ϕ1(z) and f(z)=e−βz2f1(z), where ϕ1 and f1 have genus 0 or 1 and α,β⩾0. If αβ<1/4 and ϕ has infinitely many zeros, then ϕ(D)f(z) has only simple real zeros, where D denotes differentiation.