Uniform foliation conjecture for fibers of the pure N-soliton manifold

Let $U$ be the spectral-parameter region and let $\mathbf M_U^N$ be the pure $N$-soliton set. For a spectral parameter ${\mathbf z}$, let $\mathbf M_{\mathbf z}$ denote the fiber of the spectral-parameter map on $\mathbf M_U^N$.

Uniform foliation conjecture. The fibers $\mathbf M_{\mathbf z}$ provide a uniformly smooth foliation of $\mathbf M_U^N$.

The foliation is smooth away from the complications caused by multiple eigenvalues. When all eigenvalues are simple, the fibers have a clear product-type topology, whereas multiplicities change the topology and obstruct uniform parametrizations; the conjectured uniform smooth structure remains open.