The transversal ratio conjecture for 4-polytopes

For a $d$-polytope, let the transversal ratios be denoted by $\rho_d^\mathrm{P}$ and $\rho_d^\mathrm{S}$, corresponding to polytopes and simplicial polytopes, respectively. A polytope has a transversal if a set of its vertices meets every facet. Transversal ratio conjecture. There are $4$-polytopes requiring all but a vanishingly small portion of vertices for a transversal, i.e.,

$$\rho_4^\mathrm{P}=\rho_4^\mathrm{S}=1.$$

The theorem preceding this conjecture establishes the analogous equality for dimensions $d\geq 5$. The conjecture concerns the remaining dimension $d=4$, and is motivated by constructions of $3$-dimensional spheres with unbounded chromatic number.