Packing dimension of vertical projections in the Heisenberg group

It is shown that if $A$ is a Borel subset of the first Heisenberg group, with Hausdorff dimension satisfying $2< \dim A < 3$, then the packing dimensions of vertical projections of $A$ are almost surely not less than $\dim A$, where both packing and Hausdorff dimensions are defined with respect to the Kor\'anyi metric. For the Hausdorff dimension of the projections, a weaker almost sure lower bound is obtained which improves the known bound in the range $2 < \dim A < \frac{1}{8}\left( 17 + \sqrt{33}\right) \approx 2.84$. The bound is slightly larger than $1+\frac{1}{2} \dim A$ and behaves similarly near $\dim A =2$. Both proofs rely on a variable coefficient local smoothing inequality.