Let $M$ be a smooth closed manifold and $T$ be a triangulation of $M$. Endow each simplex of $T$ with the Euclidean metric making it a regular simplex; this gives a piecewise Euclidean metric $g_0$ on $M$, which is singular on (part of) the codimension $2$ skeleton of $T$.

Is it possible to approximate $g_0$ by a smooth Riemannian metric? The approximation should in particular change length of curves and the volume by arbitrarily small amounts.

I guess the answer is positive and well-known, but I did manage to find a reference (in particular, several works ask the smoothing to satisfy certain curvature assumptions, which I do not). Is there a reference or are there obstruction to smoothing?