Conjugacy of positive-weight torus actions

Determine whether, for affine varieties X and Y over ℂ equipped with ℂ* actions having strictly positive weights and finitely many fixed points, every isomorphism X ≅ Y can be replaced by an isomorphism φ: X → Y for which Im(α) and φ^{-1}(Im(β)) are conjugate in Aut(X).

Background

The paper reformulates compatibility of two gradings in the commutative geometric setting. A nonnegative grading corresponds to a ℂ* action, and simultaneous compatibility of the gradings corresponds to conjugacy of the associated one-parameter subgroups in the automorphism group. An example involving stabilized affine surfaces motivates the question.

References

Let $X$ and $Y$ be affine varieties over $\mathbb{C}$, equipped with $\mathbb{C}\ast$-actions $\alpha \colon \mathbb{C}\ast \to \Aut(X) $ and $\beta \colon \mathbb{C}\ast \to \Aut(Y) $ with strictly positive weights and finitely many fixed points. If $X \simeq Y$, does there exist an isomorphism $\varphi \colon X \to Y$ such that $Im(\alpha)$ and $\varphi{-1}(Im(\beta))$ are conjugate in $\Aut(X)$?

— Isomorphisms of graded semiconnected algebras  (2609.03288 - Dramburg, 3 Sep 2026) in Section 4, Further questions and observations, final Question