  Using the sunflower method, we show that if $\theta \in (0,1) \cap
\mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $\theta$-intersecting
family over $[n]$, then $\lvert \mathcal{F} \rvert = O(n)$, and that if
$\mathcal{F}$ is $o(n^{1/3})$-bounded, then $\lvert \mathcal{F} \rvert \leq
(\frac{3}{2} + o(1))n$. This partially solves a conjecture raised in
(Balachandran et al., Electron J. Combin. 26 (2019), #P2.40) that any
$\theta$-intersecting family over $[n]$ has size at most linear in $n$, in the
regime where we have no very large sets.
