  Lattice QCD simulations provide crucial information about the worldsheet
dynamics of confining strings (flux tubes). An accurate extraction of the
worldsheet $S$-matrix from lattice spectra requires accounting for polarization
effects. Approximate integrability of the low energy worldsheet theory makes it
possible to apply the Thermodynamic Bethe Ansatz to incorporate polarization
effects at all orders in the number of windings and at the leading order in the
derivative expansion. However, a systematic application of this technique in
the presence of non-integrable effects and for multiparticle states becomes
increasingly challenging. We point out that a recently understood equivalence
between gravitational dressing and $T\bar{T}$ deformation provides a fully
systematic and straightforward recipe to incorporate the leading polarization
effects in the presence of an arbitrary inelasticity and for any number of
particles. We illustrate this technique with several examples.
