Extension invariance conjecture for filtered lambda-ring structures

Let $n$ be an integer with $n\geq 3$, and let $R$ and $S$ be isomorphic filtered $\lambda$-ring structures on

$$\mathbf{Z}[x]/(x^n).$$

Let $\sigma:R\xrightarrow{\cong}S$ be an isomorphism. A filtered $\lambda$-ring structure $\widetilde R$ on $\mathbf{Z}[x]/(x^{n+1})$ is an extension of $R$ when

$$\psi^p_{\widetilde R}(x)\equiv\psi^p_R(x)\pmod{x^n}$$

for every prime $p$. Extension invariance conjecture. There exists an isomorphism $\sigma:R\xrightarrow{\cong}S$ such that, for every extension $\widetilde R$ of $R$, the transported operations

$$\psi^p_{\widetilde S}(x)=(\sigma^{-1}\circ\psi^p_{\widetilde R}\circ\sigma)(x)$$

form a filtered $\lambda$-ring structure $\widetilde S$ on $\mathbf{Z}[x]/(x^{n+1})$. The statement concerns whether extensions are preserved under a suitably chosen isomorphism and remains open in the supplied text.