  Determining the neutrino mass hierarchy and measuring the CP-violating phase
$\delta_{CP}$ are two of the main aims in neutrino physics today. The upcoming
T2HK (with small matter effects and high statistics) and DUNE (with large
matter effects) experiments have been shown to have excellent sensitivity to
$\delta_{CP}$ and the neutrino mass hierarchy, respectively. The recent T2HKK
proposal aims to improve the hierarchy sensitivity of T2HK by placing one of
the two tanks of the HK detector at a site in Korea, to collect data at $\sim
1100$ km baseline. In light of the fact that DUNE will anyway collect data at
$\sim 1300$ km, we explore whether it is advantageous to collect additional
long-baseline data as proposed with T2HKK, or to enhance the
$\delta_{CP}$-precision with the `conventional' T2HK by keeping both detector
tanks in Japan. We do this by comparing the physics reach of these two options
in conjunction with DUNE. We find that DUNE+T2HKK is better at excluding the
wrong hierarchy, reaching $\Delta\chi^2 > 175$ irrespective of the true
parameters. While DUNE+T2HK can measure $\delta_{CP}$ with more precision in
some parts of the parameter space, both DUNE+T2HK and DUNE+T2HKK perform
equally well near the current best-fit point, giving a $\delta_{CP}$ width of
around $15^\circ$. The T2HKK setup allows us to correlate and constrain the
systematic errors between the two detectors collecting data from the same
source, which can increase the sensitivity of the experiment by up to 25\%.
Such a reduction of the systematic errors is crucial for determining the
oscillation parameters with greater significance.
