Helly-type intersection conjecture for projection-direction sets

Let $\mathcal{F}$ be an intersecting family of convex sets in $\mathbb{R}^3$. For each triple $T=\{F_1,F_2,F_3\}\in {\mathcal{F}\choose 3}$, let $D_T\subset S^2$ consist of the directions $\mathbf{a}$ for which the orthogonal projections of $F_1,F_2,F_3$ onto the plane $x\cdot\mathbf{a}=0$ have a common point. Intersection conjecture for the sets $D_T$. For any three triples $T_1,T_2,T_3$, one has

$$D_{T_1}\cap D_{T_2}\cap D_{T_3}\neq\emptyset.$$

Together with the surrounding proposed lemmas, this would help prove that a line pierces $c|\mathcal{F}|$ members of $\mathcal{F}$ for some constant $c$. The supplied text does not state whether the assertion is resolved.