The DMPK limiting-density conjecture for ballistic initial conditions

Let $n\to\infty$ with fixed $\mathbb{M}$. Let $\tilde{\rho}(\tilde{\lambda};\mathbb{M})$ be the limiting empirical probability density of the solution of the DMPK equation with ballistic initial condition, and set $\tilde{\lambda}=\sinh^2 x$. Define the resulting density by

$$\rho(x;\mathbb{M})=\sinh(2x)\tilde{\rho}(\sinh^2x;\mathbb{M}).$$

DMPK limiting-density conjecture. The density $\rho(x;\mathbb{M})$ is related to $\psi(x)$ by

$$\rho(x;\mathbb{M})=2x\psi(x^2),$$

where $\psi(x)$ is the function given by the paper's equation defining $\psi$ in the linear-potential case, with $V(x)=x/\mathbb{M}$.

The claim is motivated by identifying the transform $U(\zeta;\mathbb{M})=\zeta\tilde{G}(\zeta^2)=\sqrt{\mathbf{I}_2(\zeta^2)}$ with the solution of the functional equation derived from the DMPK evolution. The assumption that the limiting empirical density exists and equals $\tilde{\rho}$ is explicitly described as physically convincing but mathematically unproved, so the asserted relation remains open.