Gromov–Lees-type homotopy equivalence for polyhedral isotropic immersions

Let $\Sigma$ be a surface endowed with a triangulation $\mathscr{T}$. Let $V$ be the ambient symplectic vector space in which the immersions take values, and let $\mathscr{M}(\mathscr{T})$ denote the space of polyhedral immersions associated with $\mathscr{T}$. Up to passing to a subdivision of $\mathscr{T}$, the space of polyhedral isotropic immersions in $\mathscr{M}(\mathscr{T})$ is homotopically equivalent to the space of smooth isotropic immersions of $\Sigma$ in $V$. Gromov–Lees-type conjecture. Up to passing to a subdivision of $\mathscr{T}$, the space of polyhedral isotropic immersions in $\mathscr{M}(\mathscr{T})$ is homotopically equivalent to the space of smooth isotropic immersions of $\Sigma$ in $V$. This is proposed as a consequence of a conjectural polyhedral analogue of the Gromov–Lees theorem, motivated by the approximation scheme for isotropic immersions. The statement remains open in the supplied text.