The asymptotic robustness limit conjecture for matchings

Let $G$ range over graphs. For each $K\in\mathbb{N}$, let $\alpha_K$ denote the smallest $K$-asymptotic robustness ratio over all graphs, and let $\alpha^*(G)$ denote the randomized robustness ratio of $G$. Asymptotic robustness limit conjecture.

$$\lim_{K \rightarrow \infty} \alpha_{K} = \inf_{G} \alpha^*(G)$$

This conjecture would follow from a sparsification result for arbitrary randomized matchings, extending the constant-support sparsification established for the distributions used earlier in the paper. It asserts that the limiting deterministic asymptotic robustness ratio equals the worst-case randomized robustness ratio.