Exclusive emissions of rho-mesons and the unintegrated gluon distribution

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.