Sparse minimizer conjecture for induced subgraphs of the Kneser graph

Let $n$ be sufficiently large compared to $k$ and $s$, and let $s\leq\lambda\leq s+1$. Let $\mathcal{D}_i$ denote the $i$th star and let $\mathcal{F}_{[s+1]}^{\{i\}}$ denote the members of $\mathcal{F}$ whose intersection with $[s+1]$ is exactly $\{i\}$. A minimizer is a family $\mathcal{F}\subseteq\binom{[n]}{k}$ of size parameter $\lambda$ whose maximum degree $\Delta(\mathcal{F})$ is minimum among all families of the same size.

Sparse minimizer conjecture. There exists a minimizer $\mathcal{F}$ such that

$$\mathcal{F}\subseteq\mathcal{D}_1\cup\mathcal{D}_2\cup\cdots\cup\mathcal{D}_{s+1},$$ $$\mathcal{D}_i\cap\mathcal{D}_j\subseteq\mathcal{F}\quad\text{for all }i,j\in[s+1],$$

and

$$\left|\mathcal{F}_{[s+1]}^{\{i\}}\mathbin{\triangle}\mathcal{F}_{[s+1]}^{\{j\}}\right|\leq 1\quad\text{for all }i,j\in[s+1].$$

This is known when $\lambda=s$ is an integer, since every minimizer is then a union of $s$ stars. For nonintegral $\lambda$ in the stated range, the existence of a minimizer with these properties remains open.