Generalized skew derivations acting as Jordan homomorphism

Suppose \({\mathcal {R}}\) is a prime ring with characteristic other than two and \(\nu (s_1,\ldots , s_n)\) is a non-central multilinear polynomial over \({\mathcal {C}}\), which is non-identity. If \({\mathcal {H}}_1\) and \({\mathcal {H}}_2\) are two generalized skew derivations on the ring \({\mathcal {R}}\), satisfying the equation

$$\begin{aligned} {\mathcal {H}}_1({\mathcal {H}}_2(\nu (s)^2))={\mathcal {H}}_2(\nu (s))^2 \end{aligned}$$

for all \(s = (s_1, \ldots , s_n) \in {\mathcal {R}}^n.\) Then, we provide a comprehensive analysis of the mappings \( {\mathcal {H}}_1\) and \({\mathcal {H}}_2\) outlining their complete structure.