  One long-standing tension in the determination of neutrino parameters is the
mismatched value of the solar mass square difference, $\Delta m_{21}^2$,
measured by different experiments: the reactor antineutrino experiment KamLAND
finds a best fit larger than the one obtained with solar neutrino data. Even if
the current tension is mild ($\sim 1.5\sigma$), it is timely to explore if
independent measurements could help in either closing or reassessing this
issue. In this regard, we explore how a future supernova burst in our galaxy
could be used to determine $\Delta m_{21}^2$ at the future Hyper-Kamiokande
detector, and how this could contribute to the current situation. We study
Earth matter effects for different models of supernova neutrino spectra and
supernova orientations. We find that, if supernova neutrino data prefers the
KamLAND best fit for $\Delta m_{21}^2$, an uncertainty similar to the current
KamLAND one could be achieved. On the contrary, if it prefers the solar
neutrino data best fit, the current tension with KamLAND results could grow to
a significance larger than $5\sigma$. Furthermore, supernova neutrinos could
significantly contribute to reducing the uncertainty on $\sin^2\theta_{12}$.
