  For degree $\pm 1$ harmonic maps from $\mathbb{R}^2$ (or $\mathbb{S}^2$) to
$\mathbb{S}^2$, Bernand-Mantel, Muratov and Simon
\cite{bernand2021quantitative} recently establish a uniformly quantitative
stability estimate. Namely, for any map $u:\mathbb{R}^2\to \mathbb{S}^2$ with
degree $\pm 1$, the discrepancy of its Dirichlet energy and $4\pi$ can linearly
control the $\dot H^1$-difference of $u$ from the set of degree $\pm 1$
harmonic maps. Whether a similar estimate holds for harmonic maps with higher
degree is unknown. In this paper, we prove that a similar quantitative
stability result for higher degree is true only in local sense. Namely, given a
harmonic map, a similar estimate holds if $u$ is already sufficiently near to
it (modulo M\"{o}bius transform) and the bound in general depends on the given
harmonic map. More importantly, we investigate an example of degree 2 case
thoroughly, which shows that it fails to have a uniformly quantitative estimate
like the degree $\pm 1$ case. This phenomenon show the striking difference of
degree $\pm1$ ones and higher degree ones. Finally, we also conjecture a new
uniformly quantitative stability estimate based on our computation.
