  The positivity bound for the transverse asymmetry $A_2$ may be improved and
applied for each quark flavor separately. We use it to test the consistency of
twist-two approximation for the transverse spin-dependent ($g_T$) and the
longitudinal spin-averaged ($F_L$) structure functions. While it is satisfied
in the case of $u$ quarks, it might be violated for $d$ quarks in a region of
moderate $x$ because of its negative polarization. We attribute this
inconsistency to a stronger twist-three contribution, whose existence found by
the QCD sum rule method is a long-standing puzzle.
