The Hodge-theoretic anabelian conjecture for varieties embedded in products of hyperbolic curves

Let $X$ and $Y$ be smooth, geometrically connected varieties over

\mathbb{C}$ which can be embedded as closed subschemes into a product of hyperbolic curves over

\mathbb{C}$. Here$X($\mathbb{C})^{\sim}$ and $Y($\mathbb{C})^{\sim}$denote their schematic homotopy types, and$\mathrm{Isom}_{\mathbb{C}^}$denotes the space of$\mathbb{C}^$-equivariant isomorphisms defined by the preceding construction. Hodge-theoretic anabelian conjecture. The natural map

$$\mathrm{Isom}_{\mathbb{C}}(X,Y)\longrightarrow \mathrm{Isom}_{\mathbb{C}^*}\bigl(X(\mathbb{C})^{\sim},Y(\mathbb{C})^{\sim}\bigr)$$

admits a retraction. This is proposed as a Hodge-theoretic analogue of anabelian results for varieties related to products of hyperbolic curves; the supplied text does not establish its resolution.