Hegedüs' conjecture on skew Bollobás systems of projective subspaces

Let $W$ be an $n$-dimensional projective space over an arbitrary field $\mathbb{F}$, and let $\mathcal{P}=\{(A_i,B_i)\}_{i=1}^m$ be a skew Bollobás system of projective subspaces of $W$, meaning that

$$A_i\cap B_i=\emptyset \quad\text{for every }i\in [m],$$

and

$$A_i\cap B_j\neq\emptyset \quad\text{for all }1\leq i<j\leq m.$$

Hegedüs' conjecture. Every such system satisfies

$$m\leq 2^{\,n+1}-2.$$

Hegedüs proposed this bound in 2015 as a projective-subspace analogue of the skew Bollobás theorem. The supplied source does not state whether the conjecture has been resolved.