  Searching for Higgs bosons beyond the Standard Model (BSM) is one of the most
important missions for hadron colliders. As a landmark of BSM physics, the MSSM
Higgs sector at the LHC is expected to be tested up to the scale of the
decoupling limit of O(1) TeV, except for a wedge region centered around
$\tan\beta \sim 3 -10$, which has been known to be difficult to probe. In this
article, we present a dedicated study testing the decoupled MSSM Higgs sector,
at the LHC and a next-generation $pp$-collider, proposing to search in channels
with associated Higgs productions, with the neutral and charged Higgs further
decaying into $tt$ and $tb$, respectively. In the case of neutral Higgs we are
able to probe for the so far uncovered wedge region via $pp\to bb H/A \to
bbtt$. Additionally, we cover the the high $\tan\beta$ range with $pp\to bb H/A
\to bb\tau\tau$. The combination of these searches with channels dedicated to
the low $\tan\beta$ region, such as $pp\to H/A \to tt$ and $pp\to tt H/A \to
tttt$ potentially covers the full $\tan\beta$ range. The search for charged
Higgs has a slightly smaller sensitivity for the moderate $\tan\beta$ region,
but additionally probes for the higher and lower $\tan\beta$ regions with even
greater sensitivity, via $pp\to tb H^\pm \to tbtb$. While the LHC will be able
to probe the whole $\tan\beta$ range for Higgs masses of O(1) TeV by combining
these channels, we show that a future 100 TeV $pp$-collider has a potential to
push the sensitivity reach up to $\sim \mathcal O(10)$ TeV. In order to deal
with the novel kinematics of top quarks produced by heavy Higgs decays, the
multivariate Boosted Decision Tree (BDT) method is applied in our collider
analyses. The BDT-based tagging efficiencies of both hadronic and leptonic
top-jets, and their mutual fake rates as well as the faking rates by other jets
($h$, $Z$, $W$, $b$, etc.) are also presented.
