  We introduce a practical criterion that justifies the propagation and
appearance of $L^{p}$-norms for the solutions to the spatially homogeneous
Boltzmann equation with very soft potentials without cutoff. Such criterion
also provides a new conditional stability result for classical solutions to the
equation. All results are quantitative. Our approach is inspired by a recent
analogous result for the Landau equation derived in arXiv:2306.15729 and
generalises existing conditional results related to higher integrability
properties and stability of solutions to the Boltzmann equation with very soft
potentials.
