  We study the Higgs mass spectrum as predicted by a Maximally Symmetric Two
Higgs Doublet Model (MS-2HDM) potential based on the SO(5) group, softly broken
by bilinear Higgs mass terms. We show that the lightest Higgs sector resulting
from this MS-2HDM becomes naturally aligned with that of the Standard Model
(SM), independently of the charged Higgs boson mass and $\tan \beta$. In the
context of Type-II 2HDM, SO(5) is the simplest of the three possible symmetry
realizations of the scalar potential that can naturally lead to the SM
alignment. Nevertheless, renormalization group effects due to the hypercharge
gauge coupling $g'$ and third-generation Yukawa couplings may break sizeably
this SM alignment, along with the custodial symmetry inherited by the SO(5)
group. Using the current Higgs signal strength data from the LHC, which
disfavour large deviations from the SM alignment limit, we derive lower mass
bounds on the heavy Higgs sector as a function of $\tan\beta$, which can be
stronger than the existing limits for a wide range of parameters. In
particular, we propose a new collider signal based on the observation of four
top quarks to directly probe the heavy Higgs sector of the MS-2HDM during the
run-II phase of the LHC.
