Incompatible intersection properties

Let $\mathcal{F}\subset 2^{[n]}$ be a family in which any three sets have non-empty intersection and any two sets have at least $38$ elements in common. The nearly best possible bound $\mathrm{\mid }\mathcal{F}\mathrm{\mid }\le 2^{n-2}$ is proved. We believe that $38$ can be replaced by $3$ and provide a simple-looking conjecture that would imply this.