Generalized skew derivations acting as Jordan homomorphism

Suppose \({\mathcal {R}}\) $\mathcal{R}$ is a prime ring with characteristic other than two and \(\nu (s_1,\ldots , s_n)\) $\nu (s_1,\dots ,s_n)$ is a non-central multilinear polynomial over \({\mathcal {C}}\) $\mathcal{C}$ , which is non-identity. If \({\mathcal {H}}_1\) ${\mathcal{H}}_1$ and \({\mathcal {H}}_2\) ${\mathcal{H}}_2$ are two generalized skew derivations on the ring \({\mathcal {R}}\) $\mathcal{R}$ , satisfying the equation \(\begin{aligned} {\mathcal {H}}_1({\mathcal {H}}_2(\nu (s)^2))={\mathcal {H}}_2(\nu (s))^2 \end{aligned}\) $$\begin{array}{c}\hfill {\mathcal{H}}_1({\mathcal{H}}_2(\nu {(s)}^2))={\mathcal{H}}_2{(\nu (s))}^2\end{array}$$ for all \(s = (s_1, \ldots , s_n) \in {\mathcal {R}}^n.\) $s=(s_1,\dots ,s_n)\in {\mathcal{R}}^n.$ Then, we provide a comprehensive analysis of the mappings \( {\mathcal {H}}_1\) ${\mathcal{H}}_1$ and \({\mathcal {H}}_2\) ${\mathcal{H}}_2$ outlining their complete structure.