  A decade ago, it was shown that associated with any model for $\mathsf{U}(1)$
duality-invariant nonlinear electrodynamics there is a unique $\mathsf{U}(1)$
duality-invariant supersymmetric nonlinear sigma model formulated in terms of
chiral and complex linear superfields. Here we study the ${\cal N}=1$
superconformal $\sigma$-model analogue of the conformal duality-invariant
electrodynamics known as the ModMax theory. We derive its dual formulation in
terms of chiral superfields and show that the target space is a K\"ahler cone
with $\mathsf{U}(1)\times \mathsf{U}(1)$ being the connected component of the
isometry group.
