  We study superconformal deformations of the $T_\rho^{\hat\rho}[SU(N)]$
theories of Gaiotto-Hanany-Witten, paying special attention to mixed-branch
operators with both electrically- and magnetically-charged fields. We explain
why all marginal ${\cal N}=2$ operators of an ${\cal N}=4$ CFT$_3$ can be
extracted unambiguously from the superconformal index. Computing the index at
the appropriate order we show that the mixed moduli in
$T_\rho^{\hat\rho}[SU(N)]$ theories are double-string operators transforming in
the (Adjoint, Adjoint) representation of the electric and magnetic flavour
groups, up to some overcounting for quivers with abelian gauge nodes. We
comment on the holographic interpretation of the results, arguing in particular
that gauged supergravities can capture the entire moduli space if, in addition
to the (classical) parameters of the background solution, one takes also into
account the (quantization) moduli of boundary conditions.
