= Solution
At <linear order> a coordinate displacement $\xi^\mu$ changes a <metric perturbation> by $q_{\mu\nu}\mapsto q_{\mu\nu}-\mathcal L_\xi\bar g_{\mu\nu}$. Work in <conformal time> with $\bar g_{\mu\nu}=a^2\operatorname{diag}(-1,1,1,1)$, write $\xi^0=\alpha$ and lower spatial indices with $\delta_{ij}$. Directly evaluating the <Lie derivative of a covariant tensor field> gives
$$
q'_{00}=q_{00}+2a^2(\alpha'+\mathcal H\alpha),\qquad q'_{0i}=q_{0i}-a^2(\xi_i'-\partial_i\alpha),\qquad\mathcal H=\frac{a'}a.
$$
Here primes on $\alpha,\xi_i$ mean conformal-time derivatives, while primes on $q'_{0\mu}$ label transformed components. Setting the transformed lapse and shift perturbations to zero requires
$$
\alpha'+\mathcal H\alpha=-\frac{q_{00}}{2a^2},\qquad \xi_i'=\frac{q_{0i}}{a^2}+\partial_i\alpha.
$$
These are first-order time equations. Their explicit local solutions are
$$
\alpha(\tau,x)=\frac1{a(\tau)}\left[C(x)-\int^{\tau}\frac{q_{00}(s,x)}{2a(s)}\,ds\right],\qquad
\xi_i(\tau,x)=D_i(x)+\int^{\tau}\left[\frac{q_{0i}(s,x)}{a(s)^2}+\partial_i\alpha(s,x)\right]ds.
$$
Thus four coordinate functions can impose the four synchronous conditions in any regular perturbative patch. This is <synchronous gauge fixing by coordinate displacement>. The arbitrary spatial functions $C,D_i$ remain as residual gauge freedom; the conditions alone do not fix the coordinate system completely.
Geometrically, synchronous coordinates follow a congruence of freely falling observers launched normally from a spatial slice, using their <proper time>. This gives unit lapse and zero shift in <cosmic time>; conversion to background <conformal time> gives the displayed convention. “Always possible” is local: <geodesic> caustics or singularities can prevent a global synchronous chart. Such global obstructions do not invalidate local <linear cosmological perturbation theory>.
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