In this talk I will discuss a joint work with Yuanpu Liang in which we establish some results concerning the symplectic capacities defined by Gutt and Hutchings using $S^1$-equivariant symplectic homology. Our primary result settles a version of the recognition question in the negative. We prove that the Gutt-Hutchings capacities, together with the volume, do not constitute a complete set of symplectic invariants for star-shaped (in fact convex) domains with smooth boundary. We also prove that, even for star-shaped domains with smooth boundaries, these capacities are mutually independent and are independent from the volume. The constructions that demonstrate these independence properties​ are not exotic. They are convex and concave toric domains. The new tool used here is a significant simplification of the formulae of Gutt and Hutchings for the capacities of convex/concave toric domains, that holds under an additional symmetry assumption. This allows us to identify new mutual blind spots of the capacities which are then used to construct the desired examples.