Piecewise smooth isometric immersion conjecture for Riemannian surfaces

Let $(M,g)$ be a Riemannian surface, and let a piecewise smooth isometric immersion be a local topological embedding whose restriction to each closed face of a triangulation of $M$ is smooth and preserves the metric. A regular homotopy class is a homotopy class of immersions through immersions.

Piecewise smooth immersion conjecture. There exists a piecewise smooth isometric immersion

$$f\colon M\to \mathbb{R}^3$$

in each regular homotopy class.

The conjecture proposes a geometrically meaningful piecewise smooth analogue of the $C^1$ isometric immersion theorem for surfaces. Numerical experiments support it, while the general existence assertion remains open.