Asymptotic saturation conjecture for area-minimizing representatives modulo n

Let $M$ be the setting of the preceding discussion, let $d$ be the chain dimension, and let $g$ be a Riemannian metric. For each $n$, consider the proportion in the displayed liminf of classes in $H_d(M,\mathbb{Z}/n\mathbb{Z})$ whose all $\mathbb{Z}/n\mathbb{Z}$-area-minimizing representatives are $\mathbb{Z}$-cycles, among classes admitting $\mathbb{Z}$-representatives. Asymptotic saturation conjecture. The value of the left-hand side of the displayed liminf is $1$. The preceding result proves only that this liminf is positive; the conjecture asserts the stronger asymptotic statement that the proportion tends to full density.