Linear bound for the invariant-subspace Helly number

Let $\mathcal{A}$ be a finite family of linear operators $\mathbb K^d\to\mathbb K^d$, where $\mathbb K$ is an arbitrary field and $d\geqslant 2$. Define $HI(\mathbb K^d)$ to be the minimal integer $l$ such that if any $l$ or fewer elements of $\mathcal{A}$ have a common non-trivial invariant subspace, then all operators in $\mathcal{A}$ have a common non-trivial invariant subspace. Linear invariant-subspace Helly conjecture.

$$HI(\mathbb K^d)=O(d).$$

The preceding discussion gives the bound $HI(\mathbb K^d)\leq d^2-d+1$, and the conjecture predicts a linear improvement in the dimension. The source does not provide a proof or a sharper asymptotic bound.