  We investigate the basin of attraction properties and its boundaries for
chimera states in a circulant network of H\'enon maps. Chimera states, for
which coherent and incoherent domains coexist, emerge as a consequence of the
coexistence of basin of attractions for each state. It is known that the
coexisting basins of attraction lead to a hysteretic behaviour in the diagrams
for the density of incoherent and coherent states as a function of a varying
parameter. Consequently, the distribution of chimera states can remain
invariant by a parameter change, as well as it can suffer subtle changes when
one of the basin ceases to exist. A similar phenomenon is observed when
perturbations are applied in the initial conditions. By means of the
uncertainty exponent, we characterise the basin boundaries between the coherent
and chimera states, and between the incoherent and chimera states, and uncover
fractal and riddled boundaries, respectively. This way, we show that the
density of chimera states can be not only moderately sensitive but also highly
sensitive to initial conditions. This chimera's dilemma is a consequence of the
fractal and riddled nature of the basins boundaries.
