  Around 2007, A. Chang, J. Qing, and P. Yang proved a conformal gap theorem
for Bach-flat metrics with round sphere as the model case. In this article, we
extend this result to prove conformally invariant gap theorems for Bach-flat
$4$-manifolds with $(\mathbb{CP}^2, g_{FS})$ and
$(\mathbb{S}^2\times\mathbb{S}^2,g_{prod})$ as model cases. An iteration
argument plays an important role in the case of $(\mathbb{CP}^2, g_{FS})$ and
the convergence theory of Bach-flat metrics is of particular importance in the
case of $(\mathbb{S}^2\times\mathbb{S}^2,g_{prod})$.
