Non-maximal quasi-isometric $\mathrm{PU}_{2,n+1}$-representations via alternating surfaces in complex pseudo-hyperbolic spaces

For every admissible non-maximal Toledo invariant $t$, we construct a locus of irreducible representations of a closed genus-$g$ surface group into $\mathrm{PU}_{2,n+1}$ with Toledo invariant $t$ whose orbit maps are quasi-isometric embeddings. These loci have real codimension $10(g-1)$ or $10(g-1)+