Weak Lefschetz conjecture for balanced minimal cycle complexes

Let $\bm{a}=(a_1,\ldots,a_m)\in\mathbb{Z}_{>0}^m$ with $d=\sum_{i=1}^m a_i\geq 3$ and $\bm{a}\neq(d-1,1),(1,d-1)$. An $(d-1)$-dimensional simplicial complex $\Delta$ is $\bm{a}$-balanced if it has a map $\kappa:V(\Delta)\to[m]$ such that $|F\cap\kappa^{-1}(i)|\leq a_i$ for every face $F$ and every $i\in[m]$; such a map is an $\bm{a}$-coloring. Its Stanley–Reisner ring is $\mathbb{R}[\Delta]=\mathbb{R}[x_v:v\in V(\Delta)]/I_\Delta$, with the $\mathbb{N}^m$-grading determined by $\deg x_v=\bm{e}_{\kappa(v)}$. An $\bm{a}$-colored system of parameters is a system of parameters $\Theta=(\theta_1,\ldots,\theta_d)$ with exactly $a_i$ parameters of degree $\bm{e}_i$. Let $\Delta$ be an $\bm{a}$-balanced minimal $(d-1)$-cycle complex. Weak Lefschetz conjecture. There is an $\bm{a}$-colored system of parameters $\Theta=(\theta_1,\ldots,\theta_d)$ for $\mathbb{R}[\Delta]$ and a linear form $\omega\in\mathbb{R}[\Delta]_1$ such that

$$(\times\omega):(\mathbb{R}[\Delta]/\Theta\mathbb{R}[\Delta])_1\longrightarrow(\mathbb{R}[\Delta]/\Theta\mathbb{R}[\Delta])_2$$

is injective. This conjecture extends the result of Cook et al. for balanced simplicial $2$-spheres to balanced minimal cycle complexes, with the exceptional vectors explicitly excluded; its resolution is connected to finding infinitesimally rigid non-generic realizations of these complexes.