  We calculate the medium modification factor $I_{pA}$ for the photon-tagged
jet fragmentation functions for scenario with the quark-gluon plasma formation
in $pA$ and $pp$ collisions. We perform calculations of radiative and
collisional parton energy loss in the quark-gluon plasma with running
$\alpha_s$ which has a plateau around $Q\sim \kappa T$ with $\kappa$ fitted to
the LHC data on the heavy ion $R_{AA}$. We find that the theoretical prediction
for $I_{pA}$ in $5.02$ TeV $p$+Pb collisions are within errors consistent with
the data from ALICE [1]. However, a definite conclusion about the presence or
absence of jet quenching in $pA$ collisions cannot be drawn due to large
experimental errors of the ALICE data [1]. Our calculations show that this
requires a significantly more accurate measurement of $I_{pA}$.
