$\mathcal{N}=2$ conformal higher-spin actions in harmonic superspace

We construct linearized $4D, \mathcal{N}=2$ Weyl-type off-shell actions in harmonic superspace for the superconformal higher-spin gauge multiplets introduced in arXiv:2404.19016. This work is a followup of our arXiv:2511.16325, where an analogous construction was carried out for conformal $\mathcal{N}=2$ supergravity. We check the superconformal invariance of the new actions and their linearized gauge invariance. As the characteristic features of our formulation, the structure of the underlying prepotentials and their gauge and superconformal transformations are entirely determined by the fundamental \textit{principle of Grassmann harmonic analyticity}, while the construction of the invariant actions and of the generalized $\mathcal{N}=2$ super-Weyl tensors relies upon the \textit{principle of Grassmann half-analyticity}. This provides not only a new geometric framework for the linearized $\mathcal{N}=2$ superconformal gauge higher-spin theories, but also suggests possible ways of promoting them to the full nonlinear level. We consider a generalization to an arbitrary $\mathcal{N}=2$ supergravity background and show that the linearized spin-$\mathbf{s}$ gauge invariance, without radical modifications of the gauge transformation law and/or harmonic zero-curvature equations, is consistent only with $\mathcal{N}=2$ Weyl-flat super-backgrounds.