Density conjecture for metrics with empty-interior critical-width spectrum

Let $M$ be the closed manifold and let $\Gamma_\infty(M)$ denote the space of smooth Riemannian metrics on $M$. Let $\mathcal{M}_{ei}$ be the set of metrics for which the set of critical widths has empty interior. Density conjecture. The set $\mathcal{M}_{ei}$ is dense in $\Gamma_\infty(M)$. This is proposed as an analogue of the existence of bumpy metrics in dimensions $2\leq n\leq 6$; the statement concerns genericity of metrics whose critical-width set has empty interior, and its resolution is not supplied here.