Using the sunflower method, we show that if $θ\in (0,1) \cap \mathbb{Q}$ and $\mathcal{F}$ is a $O(n^{1/3})$-bounded $θ$-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 of Balachandran, Mathew and Mishra that any $θ$-intersecting family over $[n]$ has size at most linear in $n$, in the regime where we have no very large sets.