Conjecture on a spectral gap for the collisionless kinetic semigroup

Let $\mathsf{T}_{\mathsf{H}}$ be the generator of the $C_{0}$-semigroup $\left(U_{\mathsf{H}}(t)\right)_{t\geqslant0}$, let $\mathbf{P}_{0}$ be the spectral projection associated with the zero eigenvalue, and let $\lambda_{\beta}>0$ be the bound appearing in Proposition stated in the source. Under the assumptions of that proposition, spectral-gap conjecture. The semigroup admits a positive spectral gap $\lambda_{0}\in(0,\lambda_{\beta})$ such that

$$\left\|U_{\mathsf{H}}(t)-\mathbf{P}_{0}\right\|_{\mathscr{B}(X)}=\mathrm{O}\left(\exp\left(-\lambda_{0}t\right)\right),\qquad t\geqslant0.$$

This conjecture would upgrade convergence after projection onto the zero eigenspace from a vector-dependent estimate to a uniform operator-norm exponential decay estimate; the source does not establish it.