  The Gildener-Weinberg two-Higgs doublet model (GW-2HDM) provides a naturally
light and aligned Higgs boson, $H = H(125)$. It has been studied in the
one-loop approximation of its effective potential, $V_1$. An important
consequence is that the masses of the model's BSM Higgs bosons ($H',A,H^\pm$)
are bounded by the sum rule $\left(M_{H'}^4 + M_A^4 + 2M_{H^\pm}^4\right)^{1/4}
= 540\,{\rm GeV}$. Although they are well within reach of the LHC, searches for
them have been stymied by large QCD backgrounds. Another consequence is that
$H$ is highly aligned, i.e., $H$--$H'$ mixing is small and $H$ has only
Standard Model couplings. A corollary of this alignment is that search modes
such as $H',\,A \leftrightarrow W^+W^-,\,ZZ,\,HZ$ and $H^\pm \leftrightarrow
W^\pm Z,\,W^\pm H$ are greatly suppressed. To assess the accuracy of the sum
rule and Higgs alignment, we study this model in two loops. This calculation is
complicated by having many new contributions. We present two formulations of it
to calculate the $H$--$H'$ mass matrix, its eigenvectors $H_1,\,H_2$, and the
mass $M_{H_2}$ while fixing $M_{H_1}= 125\,{\rm GeV}$. They give similar
results, in accord with the one-loop results. Requiring $M_A = M_{H^\pm}$, we
find $180\,{\rm GeV} < M_{A,H^\pm} < 380$--$425\,{\rm GeV}$ and
$550$--$700\,{\rm GeV} > M_{H_2} > 125\,{\rm GeV}$, with $M_{H_2}$ decreasing
as $M_{A,H^\pm}$ increase. The corrections to $H$-alignment are below
${\cal{O}}(1\%)$. So, the BSM searches above will remain fruitless. Finding the
BSM Higgses requires improved sensitivity to their low masses. We discuss two
possible searches for this.
