  Three years ago, we proposed free off-shell models for ${\mathcal N}=2$
superconformal higher-spin multiplets in arbitrary conformally flat
backgrounds, constructed conserved conformal higher-spin supercurrents for a
massless hypermultiplet, and sketched the Noether procedure to generate its
cubic couplings to the superconformal higher-spin multiplets. This paper is
devoted to completing the Noether procedure. Specifically, we: (i) describe the
unique off-shell primary extensions of the conformal higher-spin supercurrents;
(ii) embed the off-shell superconformal prepotentials given in arXiv:2104.10416
into primary unconstrained isotwistor multiplets; and (iii) present the unique
gauge transformations of the hypermultiplet and the isotwistor prepotentials.
An extension of the Noether procedure beyond the cubic level is also sketched,
following the earlier ${\mathcal N}=1$ superconformal approach of
arXiv:2208.07783. Our construction is based on making use of the polar
hypermultiplet within the projective-superspace setting.
