  We provide a simpler proof of the hard Lefschetz Theorem for face rings of PL
spheres: While the algebraic theory remains the same, we replace the geometric
constructions by Pachner's Theorem. This simplifies the reasoning for an
important special case of the main result of the first author in
arxiv:1812.10454, and already implies the McMullen g-conjecture for PL spheres,
as well as the Gr\"unbaum-Kalai-Sarkaria conjecture.
