  Two-photon exchange (TPE) contributions to elastic electron-proton scattering
in the forward regime are considered. The imaginary part of TPE amplitude in
these kinematics is related to the DIS nucleon structure functions. The real
part of the TPE amplitude is obtained from the imaginary part by means of
dispersion relations. We demonstrate that the dispersion integrals for the
relevant elastic $ep$-scattering amplitude converge and do not need
subtraction. This allows us to make clean prediction for the real part of the
TPE amplitude at forward angles. We furthermore compare $e^+p$ and $e^-p$ cross
sections which depends on the real part of TPE amplitude, and predict the
positron cross section to exceed the electron one by a few per cent, with the
difference ranging from 1.4% to 2.8% for electron $lab$ energies in the range
from 3 to 45 GeV. We furthermore predict that the absolute value of this
asymmetry grows with energy, which makes it promising for experimental tests.
