The cotangent-bundle isomorphism conjecture for smooth affine varieties

Let $X$ and $Y$ be smooth affine varieties over $\mathbb{C}$, and let $T^*(X)$ and $T^*(Y)$ denote their cotangent bundles. The cotangent-bundle isomorphism conjecture. If

$$T^*(X)\cong T^*(Y),$$

then

$$X\cong Y.$$

The conjecture is motivated by the theorem in the source that Morita-equivalent rings $D(X)$ and $D(Y)$ have isomorphic cotangent bundles as symplectic varieties; whether the cotangent-bundle isomorphism determines the original smooth affine variety remains open in the supplied text.