  We develop a consistent theory of the interlayer exciton-polaron formed in
atomically-thin bilayers. Coulomb attraction between an electron and a hole
situated in the different layers results in their flexural deformation and
provides an efficient mechanism of the exciton coupling with flexural phonons.
We study the effect of layers tension on the polaron binding energy and
effective mass leading to suppression of polaron formation by the tension both
in the weak and strong coupling regimes. We also consider the role of the
nonlinearity related to the interaction between the out- and in-plane lattice
displacements and obtain the criterion of the layer sticking, where the exciton
collapses, due to the Coulomb attraction between the charge carriers.
