  We investigate a possibility that the $\mu^-\to e^+$ conversion is discovered
prior to the $\mu^-\to e^-$ conversion, and its implications to the new physics
search. We focus on the specific model including the mixing of the $SU(2)_L$
doublet- and singlet-type scalar leptoquarks, which induces not only the lepton
flavor violation but also the lepton number violation. Such a structure is
motivated by R-parity violating (RPV) supersymmetric models, where a sbottom
mediates the conversion processes. We formulate the $\mu^-\to e^+$ rate in
analogy with the muon capture in a muonic atom, and numerically evaluate it
using several target nuclei. The lepton flavor universality test of pion decay
directly limits the $\mu^-\to e^+$ rate, and the maximally allowed $\mu^-\to
e^+$ branching ratio is $\sim 10^{-18}$ under the various bounds on RPV
parameters. We show that either $\mu^-\to e^-$ or $\mu^-\to e^+$ signals can be
discovered in near future experiments. This indicates that parallel searches
for these conversions will give us significant information on the pattern of
coupling constants.
