Cut locus smeariness conjecture for real projective spaces

Let $m\geq 2$, and let $F$ and $G$ denote the Fréchet functions associated with the intrinsic and comparison distance settings, respectively, near the origin $0$ in the real projective space $\mathbb{R}P^m$. Cut locus smeariness conjecture. For every neighborhood $U$ of $0$, there is an $x\in U$ such that

$$F(x)\neq G(x).$$

This would indicate that $\mathbb{R}P^m$ admits cut locus smeariness. The statement is motivated by the fact that cut locus smeariness can occur when the cut locus has dimension $m-1$, but its resolution is not supplied here.