  A qualitative difference in the running sum for the nuclear matrix element of
the two-neutrino double-$\beta$ decay of $^{136}$Xe was found four years ago
between quasiparticle random-phase approximation (QRPA) and shell model
calculations. The former result has large increase and decrease with respect to
the excitation energy of the intermediate state, and the latter one is an
almost monotonically and mildly increasing function. My QRPA calculations
independently of the above one do not have a remarkable decrease. This
discrepancy is a serious problem affecting the reliability of calculations of
the neutrinoless double-$\beta$ decay, and the cause was unknown. I perform
several relevant test calculations and make an analytical consideration to find
the cause, which is found to be in the strength of the attractive interactions.
The possible major local decrease in the running sum is also explained
analytically. The interactions of my QRPA calculation are appropriate in terms
of the strength, thus the almost monotonic behavior is reasonable.
