Isometric standard cubes in geometric realizations of precubical sets

Let $C$ be a finite-dimensional geometric precubical set, let $n\in\mathbb{N}$, let $c\in C(n)$, and let $I^n$ be the standard $n$-cube. Write $\greal{C}$ for the geometric realization of $C$ and $\iota_c:I^n\to\greal{C}$ for the canonical inclusion. Isometric-cube conjecture. The morphism

$$\iota_c:I^n\to\greal{C}$$

is an isometry. This is intended as a global strengthening of the local isometry property of geometric realizations: paths leaving a cube should not shorten distances between points inside it.