  We present our new Atacama Large Millimeter/submillimeter Array (ALMA)
observations targeting CO(6-5) emission from three luminous Lyman break
galaxies (LBGs) at $z_{\rm spec} = 6.0293$-$6.2037$ found in the Subaru/Hyper
Suprime-Cam survey, whose [OIII]$88\mu$m and [CII]$158\mu$m emission have been
detected with ALMA. We find a marginal detection of the CO(6-5) line from one
of our LBGs, J0235-0532, at the $\simeq 4 \sigma$ significance level and obtain
upper limits for the other two LBGs, J1211-0118 and J0217-0208. Our $z=6$
luminous LBGs are consistent with the previously found correlation between the
CO luminosity and the infrared luminosity. The unique ensemble of the multiple
far-infrared emission lines and underlying continuum fed to a photodissociation
region model reveal that J0235-0532 has a relatively high hydrogen nucleus
density that is comparable to those of low-$z$ (U)LIRGs, quasars, and Galactic
star-forming regions with high $n_{\rm H}$ values, while the other two LBGs
have lower $n_{\rm H}$ consistent with local star-forming galaxies. By
carefully taking account of various uncertainties, we obtain total gas mass and
gas surface density constraints from their CO luminosity measurements. We find
that J0235-0532 locates below the Kennicutt-Schmidt (KS) relation, comparable
to the previously CO(2-1) detected $z=5.7$ LBG, HZ10. Combined with previous
results for dusty starbursts at similar redshifts, the KS relation at $z=5$-$6$
is on average consistent with the local one.
