Existence of monomial ideals with the strong Lefschetz property

Let $R=k[x_1,\dots,x_n]$ where $k$ is a field of characteristic zero. Fix $n,d\geq 3$ and let $\mu$ lie in the interval $\left[n,\binom{n+d-1}{d}\right]$. A monomial ideal $I$ minimally generated by $\mu$ monomials of degree $d$ is an ideal for which $R/I$ has the strong Lefschetz property (SLP). Existence conjecture. For every such $n$, $d$, and $\mu$, there is a monomial ideal $I$ minimally generated by $\mu$ monomials of degree $d$ such that $R/I$ has the SLP. The conjecture proposes that every admissible number of degree-$d$ minimal generators can occur for a monomial quotient with the SLP; the authors report that computational experiments support it, but no proof or resolution is given here.