= Solution
In the thin-sheet regime, the gas <scale height> is $h_g(R)=\sigma_g^2(R)/(2\pi G\Sigma_*(R))$. Hence radius-independent gas <velocity dispersion> gives
$$
\boxed{\frac{h_g(R)}{h_g(R_0)}=\frac{\Sigma_*(R_0)}{\Sigma_*(R)}}.
$$
For an <exponential galactic disk>, $\Sigma_*(R)=\Sigma_{*,0}e^{-R/R_d}$, so
$$
\boxed{h_g(R)=\frac{\sigma_g^2}{2\pi G\Sigma_{*,0}}e^{R/R_d}}.
$$
The gas therefore exhibits <galactic disk flaring> as stellar <surface density> falls outward. Stellar <galactic disks>, by contrast, commonly have roughly constant thickness over their main optical region; applying the same <scale height> relation to the stars requires $\sigma_{z,*}^2(R)\propto\Sigma_*(R)$, or $\sigma_{z,*}\propto e^{-R/(2R_d)}$ for an <exponential galactic disk>. This comparison does not follow by imposing a radially constant stellar <velocity dispersion>: that extra assumption would predict stellar <galactic disk flaring> too. The constancy used in the vertical calculation is constancy with height locally, not a mandatory constancy with radius.
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