<p>In the present work, we investigate the regularity estimates for the solutions to the non-cutoff Boltzmann equation with soft potentials. We restrict our attention to the so-called “typical rough and slowly decaying data”, which are constructed to satisfy the typical properties: low regularity and having exact polynomial decay in high velocity regimes. By exploring the degenerate and non-local properties of the collision operator, we demonstrate that (i) such data induce only finite smoothing effects for weak solutions in Sobolev spaces; (ii) this finite smoothing property implies that the Leibniz rule does not hold for high-order derivatives of the collision operator (even in the weak sense). These facts present big obstacles to proving the conjecture that solutions to the equation will become infinitely smooth instantly for both spatial and velocity variables at any positive time if the initial data have only polynomial decay in high velocity regimes.</p>