Exponential bound for near-sunflower-free uniform families

Let a near-sunflower of size $r$ be the configuration defined in the paper, and let $\mathcal{F}$ be a family of $k$-element sets. Near-sunflower exponential-bound conjecture. For $r\ge4$, if $\mathcal{F}$ contains no near-sunflower of size $r$, then

$$|\mathcal{F}|\le C^k,$$

where $C$ is a constant depending only on $r$. This would improve the general upper bound inherited from the sunflower problem, whose best stated bound is of order $(\log k)^{(1+o(1))k}$; the conjecture is presented as an open question.