= Solution
Let $U$ be the quasi-local unitary carrying the ground space of $H_0$ to that of a topologically equivalent $H$. Define
$$
\gamma_1=Uc_1U^\dagger,
\qquad
\gamma_{2M}=Uc_{2M}U^\dagger.
$$
Unitary conjugation preserves self-adjointness and the Majorana algebra. Quasi-locality spreads each endpoint operator only into an exponentially decaying tail, and the two operators preserve the ground space because $c_1,c_{2M}$ preserve the ground space of $H_0$.
Diagonalize the quadratic Hamiltonian as
$$
H=\frac12\sum_j\epsilon_j i\chi_{2j-1}\chi_{2j}.
$$
The hint gives a real expansion $\gamma=\sum_jg_j\chi_j$. Any coefficient along a pair with $\epsilon_j>0$ would create or remove a positive-energy quasiparticle and send some ground state outside the two-lowest-state subspace. Ground-space preservation therefore forces $\gamma_1$ and $\gamma_{2M}$ to have support only in the zero-energy Majorana subspace. Hence
$$
\boxed{[H,\gamma_1]=[H,\gamma_{2M}]=0}
$$
in the ideal infinite-chain limit, with only exponentially small finite-size corrections. They are the exponentially localized endpoint <Majorana zero modes> throughout the topological phase.
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