On associated primes of initial ideals

Let R be a polynomial ring over a field with r variables. Let P be a homogeneous ideal of R such that all the variables of R are not zero divisors mod P. Assume that the initial ideal of P is strongly stable. It is proven that if the irrelevant ideal is an associated prime ideal of the initial ideal of P, then the ideal generated by the first r-1 variables is also an associated prime ideal of this initial ideal.