  In presence of non standard interactions, the solar neutrino data is
consistent with two solutions, one close to the standard LMA solution with
$\sin^2\theta_{12} \simeq 0.3$ and another with $\sin^2\theta_{12}^D \simeq
0.7(=\cos^2\theta_{12})$. The latter has been called the Dark LMA (DLMA)
solution in the literature. This issue is hard to resolve via oscillations
because of the existence of the so-called "generalised mass hierarchy"
degeneracy of the neutrino mass matrix in presence of NSI. However, if the mass
hierarchy is independently determined in a non-oscillation experiment such as
neutrino-less double beta decay, the invariance of neutrino oscillation
probabilities under $\sin^2\theta_{12} \leftrightarrow \cos^2\theta_{12}$ is
lost and the possibility of resolving the LMA vs DLMA opens up. We point out
that the $P_{\mu\mu}$ channel can distinguish $\theta_{12}$ from
$\theta_{12}^D$ and study the corresponding difference in long-baseline
experiments. We show that a key ingredient required is the input from the
$P_{ee}$ channel measured at a reactor experiment. We find that if the mass
hierarchy is determined by neutrino-less double beta decay, then a combined
measurement of the effective mass squared differences in long-baseline
experiments such as T2HK and DUNE and reactor experiment such as JUNO can
resolve the DLMA conundrum to better than $3\sigma$ within 1 year for T2HK and
little more than 3 years for DUNE.
