Do closed-ultrafilter-limits keep cov(N) small?

Determine whether finite-support iterations of ccc forcings whose iterands have closed-ultrafilter-limits (c-uf-limits) can keep the covering number of the null ideal cov(N) small; equivalently, ascertain whether c-uf-limit methods provide the "inside" preservation needed to force C_{[A]^{<θ} ≤_T Cn and thus ensure cov(N) ≤ θ in the resulting model.

Background

The paper introduces closed-ultrafilter-limits (c-uf-limits) and proves they keep the evasion number e small along suitable iterations. Many left-side invariants in Cichoń's diagram are either below e or above e_ubd, so they are either automatically controlled by keeping e small or are out of reach of c-uf-limits.

The author notes that a plausible remaining candidate on the left side is the covering number of the null ideal cov(N). However, it is not clear whether c-uf-limits can keep this invariant small, and even if they could, existing forcings with c-uf-limits are typically σ-centered or subrandom, which already keep cov(N) small by other means.