Poincaré series of modules over compressed Gorenstein local rings

Given two positive integers e and s we consider Gorenstein Artinian local rings R   whose maximal ideal mm satisfies ms≠0=ms+1ms≠0=ms+1 and rankR/m(m/m2)=erankR/m(m/m2)=e. We say that R is a compressed Gorenstein local ring   when it has maximal length among such rings. It is known that generic Gorenstein Artinian algebras are compressed. If s≠3s≠3, we prove that the Poincaré series of all finitely generated modules over a compressed Gorenstein local ring are rational, sharing a common denominator. A formula for the denominator is given. When s is even this formula depends only on the integers e and s  . Note that for s=3s=3 examples of compressed Gorenstein local rings with transcendental Poincaré series exist, due to Bøgvad.