Let $X,Y$ be two affine $\mathbb{C}$-schemes. Assume that I have an isomorphism of the underlying reduced spaces:
$$f : X_{red} \stackrel{\sim}\longrightarrow Y_{red}$$ which induces an ismorphism of $\mathcal{O}_{X_{red}}$-module:

$$ f^{*} \Omega_{Y/\mathbb{C}} \longrightarrow \Omega_{X/\mathbb{C}}.$$

Can I lift $f$ to an actual isomorphism $X \stackrel{\sim}\longrightarrow Y$? Note that in the situation I am interested $Y$ is very singular. If $Y$ were smooth, I guess the theory of infinitesimal liftings would give the answer.

 This is related to this [question][1]. Indeed, the existence of such a lift would guarantee the isomorphism between the rings introduced by the OP in that other question. 


  [1]: https://mathoverflow.net/questions/354193/are-the-rings-mathbbcx-langle-x2-c-mathrmtrx-x-rangle-isomorphic/354225#354225