  Exclusive emissions of vector mesons in forward directions of rapidity offer
us a faultless chance to probe the proton structure at small-$x$. A high-energy
factorization formula is established within BFKL, given as the convolution of
an impact factor depicting the forward-meson emission and of an unintegrated
gluon distribution (UGD) driving the gluon evolution at small-$x$. As a
nonperturbative quantity, the UGD is not well known and several models for it
exist. We present recent progresses on the study of the exclusive forward
$\rho$-meson leptoproduction at HERA and EIC energies, showing how osbervables
sensitive to different polarization states of the $\rho$-particle act as
discriminators for the existing UGD models.
