Here a hexagonal 2-torus T_H means any Riemannian flat torus obtained by identifying opposite edges of a regular hexagon in the plane.

It's easy to see that T_H can be smoothly (C^∞) isometrically embedded in ℝ⁶, and for that matter (if suitably scaled) in the unit sphere S⁵, and also in ℂℙ².

But is it known whether T_H can be smoothly isometrically embedded in ℝ⁵ or ℝ⁴ ?

(Also of interest is whether T_H can be smoothly isometrically immersed in ℝ⁵ or ℝ⁴.)