The algebraic g-conjecture for homology spheres

Let $\Delta$ be a homology sphere. Let $k[\Delta]$ be its Stanley–Reisner ring, let $\Theta$ be a linear system of parameters, and define $L(\Delta)$ to be the set of pairs $(\omega,\Theta)$ for which $\omega$ is a Lefschetz element, meaning that multiplication

$$\omega^{d-2i}:(k[\Delta]/\langle\Theta\rangle)_i\to(k[\Delta]/\langle\Theta\rangle)_{d-i}$$

is an isomorphism for every $i\le\lfloor d/2\rfloor$. The algebraic g-conjecture. If $\Delta$ is a homology sphere, then

$$L(\Delta)\ne\emptyset.$$

A Lefschetz element would imply the monotonicity conditions and the M-vector condition for the g-vector, and hence the usual g-conjecture for homology spheres. The source presents this as open.