= Solution
Regard a chord as an oriented curve modulo increasing reparameterization, and write $[\gamma]$ for that class. Scale invariance of $\mu$ means that $[r\gamma]$ has the same law for each $r>0$. The <locality of a chord law under conformal changes of neighborhoods> is the following stopped-curve property: if $N,N^*$ are neighborhoods of the starting boundary point and $\Phi:N\to N^*$ is a boundary-preserving <conformal isomorphism> with $\Phi(0)=0$, then
$$
\boxed{\left[\Phi(\gamma\text{ stopped on leaving }N)\right]
\overset{\rm law}=
\left[\gamma\text{ stopped on leaving }N^*\right].}
$$
The two stopped curves are sampled from the same original law $\mu$; equality is for their unparameterized initial segments. A normalization of $\Phi'(0)$ can be imposed in an equivalent formulation, since scale invariance removes the positive dilation. In a hull-removal formulation, the <Locality property of SLE> says that a modification away from the starting point leaves the initial law unchanged until the curve reaches the modified part of the domain.
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