= Solution
For sufficiently weak local fields the bulk gap remains open, so <quasi-adiabatic continuation> supplies a quasi-local unitary $U$ mapping the unperturbed ground space to the span of the $2^k$ lowest eigenstates:
$$
|\varphi_\alpha\rangle=U|\psi_\alpha\rangle.
$$
If $S_j$, $j=1,\ldots,n-k$, are independent original stabilizer generators, define
$$
\boxed{\widetilde S_j=US_jU^\dagger.}
$$
Then $\widetilde S_j|\varphi_\alpha\rangle=|\varphi_\alpha\rangle$, and independence is preserved by conjugation. The operators $\widetilde S_j$ are quasilocal, with exponentially decaying tails, but a generic perturbation makes them non-Pauli; this is the <dressed stabilizer under a weak local perturbation>.
Exact error correction is transported with the code: the exactly correctable dressed errors are $UE_aU^\dagger$. A bare Pauli error is generally not one of these dressed operators. Its expansion in the dressed algebra has exponentially small long-range components that can act within the logical space, so
$$
\Pi_H E_a^\dagger E_b\Pi_H
=c_{ab}\Pi_H+\text{exponentially small logical terms}.
$$
Thus a set of bare Pauli errors satisfies the KL conditions only approximately, with deviations suppressed exponentially by the code distance relative to the dressing length, except at specially tuned points.
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