  We prove an analogue of the 4-dimensional local Viterbo conjecture for the
higher Ekeland-Hofer capacities: on the space of 4-dimensional smooth
star-shaped domains of unitary volume, endowed with the $C^3$ topology, the
local maximizers of the $k$-th Ekeland-Hofer capacities are those domains
symplectomorphic to suitable rational ellipsoids.
