Solution
= Solution
The <Hodge star operator> is defined by
$$
\alpha\wedge *_g\beta
=\langle\alpha,\beta\rangle_g\,\operatorname{vol}_g
$$
for forms of the same degree. In four dimensions, under $\widehat g=e^{2\psi}g$, the inner product on $p$-forms scales by $e^{-2p\psi}$ and the volume form scales by $e^{4\psi}$. Hence
$$
*_{\widehat g}\big|_{\Omega^p}
=e^{(4-2p)\psi}*_g.
$$
For $p=2$ the exponent vanishes, so the Hodge star on two-forms is conformally invariant.