Radial projections along chains

We state strong Marstrand properties for two related families of fractals in Heisenberg groups ${\mathcal{H}}^d$: limit sets of Schottky groups in good position, and attractors of self-similar IFS enjoying the open set condition in the quotient ${\mathcal{H}}^d\mathrm{/}Z$. For such a fractal $X$, we show that the dimension of ${\pi }_{x}X$ does not depend on $x\in {\mathcal{H}}^d$, where ${\pi }_x$ denotes the radial projection along chains passing through $x$. This follows from a local entropy averages argument due to Hochman and Shmerkin.