= Solution
Because density is a <scalar field>, $\widetilde\rho(\widetilde x)=\rho(x)$. Write both sides as background plus perturbation and expand the background time shift:
$$
\bar\rho(\tau)+\delta\rho(\tau,\mathbf x)
=\bar\rho(\tau+T)+\widetilde{\delta\rho}(\tau+T,\mathbf x+\nabla L)
=\bar\rho(\tau)+\bar\rho'T+\widetilde{\delta\rho}(\tau,\mathbf x).
$$
The coordinate shift acting on an already first-order perturbation contributes only at second order. Thus the <density perturbation gauge transformation> is
$$
\boxed{\widetilde{\delta\rho}=\delta\rho-\bar\rho'T.}
$$
The <uniform-density curvature perturbation> in the paper's sign convention is
$$
\zeta=-C+\frac13\nabla^2E+\mathcal H\frac{\delta\rho}{\bar\rho'}.
$$
The derivative on $\bar\rho'$ is essential and is present in the PDF; the TeX transcription drops it. The scalar-curvature combination transforms as
$$
-\widetilde C+\frac13\nabla^2\widetilde E
=-C+\frac13\nabla^2E+\mathcal HT,
$$
while the density term changes by $-\mathcal HT$. Hence \b[the two time-slicing changes cancel]:
$$
\boxed{\widetilde\zeta=\zeta.}
$$
On a uniform-density slice, $\delta\rho=0$ and this variable is the signed spatial-curvature perturbation. The construction assumes $\bar\rho'\ne0$; a pure constant-density cosmological constant does not define such a time slicing.
In <Newtonian gauge in cosmology>, $E=B=0$ and $C=-\Phi$. The background <cosmological perfect-fluid continuity equation> gives $\bar\rho'=-3\mathcal H(\bar\rho+\bar P)$, so, writing $D=\bar\rho+\bar P$,
$$
\zeta=\Phi-\frac{\delta\rho}{3D}.
$$
Set the <cosmological adiabatic sound speed> $c_a^2=\bar P'/\bar\rho'$. Then $D'=-3\mathcal H D(1+c_a^2)$. Differentiate the previous expression and insert the perturbed energy-conservation equation:
$$
\begin{aligned}
\zeta'&=\Phi'-\frac{\delta\rho'}{3D}
+\frac{\delta\rho D'}{3D^2}\\
&=\frac{\mathcal H}{D}(\delta\rho+\delta P)
+\frac{\nabla\cdot\mathbf q}{3D}
-\frac{\mathcal H}{D}(1+c_a^2)\delta\rho.
\end{aligned}
$$
Consequently the \b[curvature evolution equation in this sign convention] is
$$
\boxed{\zeta'=\frac{\mathcal H}{\bar\rho+\bar P}\delta P_{\rm nad}
+\frac{\nabla\cdot\mathbf q}{3(\bar\rho+\bar P)}
=\frac{\mathcal H}{\bar\rho+\bar P}\delta P_{\rm nad}
+\frac13\nabla\cdot\mathbf v.}
$$
The last equality defines the total energy-frame velocity by $\mathbf q=D\mathbf v$. For <adiabatic cosmological perturbations>, the <non-adiabatic pressure perturbation> vanishes. On <superhorizon scales>, the stated suppression of the velocity divergence makes the remaining gradient term negligible. Thus \b[$\zeta$ is conserved to leading order in $k/\mathcal H$]. The positive sign of the pressure source follows from the paper's definition of $\zeta$ and must not be replaced by a formula using the opposite curvature convention.
<Superhorizon conservation of uniform-density curvature> lets one carry a <primordial perturbation> from its inflationary generation through otherwise complicated eras and predict the initial conditions for later <Cosmic microwave background anisotropy> and structure growth. It relies on adiabaticity: an <isocurvature perturbation> can source curvature evolution, so the same conclusion is not automatic in an arbitrary multifield model.
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