The viscous central-invariant conjecture for integrable viscous conservation laws

Let two integrable viscous scalar conservation laws be given, with $a(u)$ denoting the coefficient of the first-order viscous term and hence their viscous central invariant. A Miura transformation is a perturbation of the identity of the form

$$u\to v=u+\sum_k\epsilon^k M_k(u_x,u_{xx},\dots).$$

Viscous central-invariant conjecture. Two integrable viscous conservation laws admitting the same viscous central invariant $a(u)$ are Miura equivalent.

This is the main conjectural classification statement for scalar viscous conservation laws, extending the finite-order result that the quasilinear part is determined by $a(u)$. The source does not report a proof or disproof.