  The isosinglet combination of the chiral-odd twist-3 distribution function
$e^u(x)+e^d(x)$ of the nucleon has outstanding properties that its first moment
is proportional to the well-known $\pi N$ sigma-term and that it contains a
$\delta$-function singularity at $x=0$. These two features are inseparably
connected in that the above sum rule would be violated, if there is no such a
singularity in $e^u(x)+e^d(x)$. Very recently, we found that the physical
origin of this $\delta$-function singularity can be traced back to the
long-range quark-quark correlation of scalar type, which signals the
spontaneous chiral symmetry breaking of the QCD vacuum. The main purpose of the
present paper is to give complete theoretical predictions for the chiral-odd
twist-3 distribution function $e^a(x)$ of each flavor $a$ on the basis of the
chiral quark soliton model, without recourse to the derivative expansion type
approximation. These theoretical predictions are then compared with the
empirical information extracted from the CLAS data of the semi-inclusive DIS
processes by assuming the Collins mechanism only. A good agreement with the
CLAS data is indicative of a sizable violation of the $\pi N$ sigma-term sum
rule, or equivalently, the existence of a $\delta$-function singularity in
$e^u(x) + e^d(x)$.
