Asymptotic unit hitting time conjecture for primitive geometric lattices

Let $L^{(n)}$ be a sequence of primitive geometric lattices, meaning geometric lattices that cannot be written as products of smaller geometric lattices, and suppose that

$$\lvert L^{(n)}\rvert\to\infty\quad\text{as }n\to\infty.$$

Let $\mathcal E(L^{(n)})$ denote the expected number of steps for the Ungarian Markov chain on $L^{(n)}$ to reach its bottom element from its top element. Primitive geometric lattice conjecture. Then

$$\mathcal E(L^{(n)})=1+o(1).$$

Geometric lattices are precisely the lattices of flats of matroids, and primitive geometric lattices correspond to connected matroids. The source proposes this asymptotic behavior as a future direction and gives no resolution.