The higher-dimensional contact-star union conjecture

Let $\mathbb P^n$ be projective $n$-space, and let $\mathbb{S}(\ell_1,\ldots,\ell_r)$ and $\mathbb{S}(m_1,\ldots,m_s)$ denote contact star configurations formed from distinct hyperplanes. Set

$$X:=\mathbb{S}(\ell_1,\ldots,\ell_r),\qquad Y:=\mathbb{S}(m_1,\ldots,m_s),$$

where $r\ge s$. The higher-dimensional contact-star conjecture. The $h$-vector of $X\cup Y$ is

$$h_{X\cup Y}=\left(1, \binom{n}{n-1}, \ldots, \binom{r-1}{n-1}, \binom{s-1}{n-1}, \ldots, \binom{n}{n-1}, 1\right).$$

In particular, if $s=r$ or $s=r-1$, then $X\cup Y$ is a Gorenstein set of points. The authors present this as a possible extension of their results to higher-dimensional spaces, based on computations; no resolution is given in the supplied source.