Vertical projections in the Heisenberg group for sets of dimension greater than 3

It is shown that vertical projections in the Heisenberg group of sets of dimension strictly greater than 3 almost surely have positive area. The proof uses the point-plate incidence method introduced by Fässler and Orponen, and also uses a similar approach to a recent maximal inequality of Zahl for fractal families of tubes. It relies on the endpoint trilinear Kakeya inequality in ${\mathbb{R}}^3$. Some related results are given on generic intersections with horizontal lines.