Exponential convergence conjecture for the sixth-order quantum-diffusion equation

Let $u$ be one of the weak solutions to the sixth-order quantum-diffusion equation constructed in the proof of the existence theorem, let $u^0$ be its initial datum, and let $U_\lambda$ denote the Gaussian equilibrium. The distance between $u(t,\cdot)$ and $U_\lambda$ is measured in $L^1(\mathbb R^d)$. Exponential convergence conjecture. There exists a constant $C(u^0)$ depending only on $u^0$ such that

$$\|u(t,\cdot)-U_\lambda\|_{L^1(\mathbb R^d)}\leq C(u^0)e^{-\lambda^3t}\qquad\text{for all }t\geq 0.$$

This conjecture asserts that the spectral gap $2\lambda^3$ of the linearization determines a global exponential convergence rate, here stated as the rate $\lambda^3$ in $L^1$. The source presents it as a consequence suggested by known nonlinear exponential stability results for the DLSS equation; no resolution is given here.