Rigidity conjecture for positively Ricci-curved Bryant-asymptotic steady solitons

Let $(M^n,g)$ be a steady gradient Ricci soliton with positive Ricci curvature. Suppose there are a ball $B_R(0)\subset\mathbb R^n$ and a diffeomorphism from $(\mathbb R^n\setminus B_R,g_0)$ to $(M,g)$ whose pullback metric satisfies the stated $C^{2,\tau}$ asymptotic behavior to the $n$-dimensional Bryant Ricci soliton $(\mathbb R^n,g_0)$, for $\tau\in(0,1)$. Rigidity conjecture. The soliton must be isometric to $(\mathbb R^n,g_0)$. This is proposed as a weakening of the paper's curvature-pinching hypothesis; the source gives no resolution, while related rigidity results are known for shrinking and expanding gradient Ricci solitons.