Weak Lefschetz property and stellar subdivisions of Gorenstein complexes

Assume sigma is a face of a Gorenstein* simplicial complex D, and k is an infinite field. We investigate the question of whether the Weak Lefschetz Property of the Stanley-Reisner ring k[D] is equivalent to the same property of the Stanley-Reisner ring k[D-sigma] of the stellar subdivision D-sigma. We prove that this is the case if the dimension of sigma is big compared to the codimension.