Holroyd–Johnson conjecture for intersecting k-separated sets

Let $[n]^{(r)}_k$ denote the collection of $k$-separated $r$-sets in $[n]^{(r)}$: sets $A=\{a_1,\dots,a_r\}$ with $a_1<a_2<\cdots<a_r$ satisfying $a_{i+1}-a_i>k$ for $i=1,\dots,r$, where $a_{r+1}=a_1+n$. Let $n$, $k$, and $r$ be positive integers satisfying

$$n\geq (k+1)r.$$

For an intersecting family $\mathcal{A}\subseteq[n]^{(r)}_k$, define the star at $1$ by

$$\mathcal{A}_1^*=\{A\in[n]^{(r)}_k:1\in A\}.$$

Holroyd and Johnson's conjecture. If $\mathcal{A}\subseteq[n]^{(r)}_k$ is intersecting, then

$$|\mathcal{A}|\leq|\mathcal{A}_1^*|.$$

The conjecture asserts an Erdős–Ko–Rado-type bound for intersecting families of $k$-separated sets. The supplied text does not state whether it has been resolved; the status is therefore recorded as open.