Asymptotic negligibility of smooth Dressian cells

Fix $r\geq 4$. Let $U(r,n)$ be the uniform matroid of rank $r$ on an $n$-element ground set, and let $\mathscr{D}(U(r,n))$ be its Dressian. For a valuation $\nu\in\mathscr{D}(U(r,n))$, let $D(\nu)$ be its cell. A valuation is smooth if it cannot be written as $\nu=\nu'+\sigma$ with $\nu',\sigma\in\mathscr{D}(U(r,n))$ where $\sigma$ is a spike, meaning that $\sigma$ has exactly one nonzero coordinate. Asymptotic negligibility conjecture for smooth cells. For each fixed $r\geq 4$,

$$\frac{\ln\#\{D(\nu):\nu\in\mathscr{D}(U(r,n)),\ \nu\text{ smooth}\}}{\ln\#\mathcal{D}(U(r,n))}\longrightarrow 0\quad\text{as }n\longrightarrow\infty.$$

The total number of Dressian cells has a lower bound from sparse paving matroids and an upper bound of order $\binom{n}{r}\log(n)^2/n$ on the logarithmic scale, while the source notes a remaining gap even in fixed rank. The conjecture predicts that smooth cells contribute negligibly to the logarithmic count, so that nonsmooth valuations predominate for uniform matroids.