Attainment and classification of the counterexample cardinalities

Determine whether the lower bounds $\operatorname{cov}$ and $\mathfrak d$ are attained in ZFC for the corresponding minima, whether the minima for posets and complete bounded distributive lattices can differ, and whether these minima coincide with classical cardinal characteristics of the continuum.

Background

The paper defines four minima measuring the least size of a poset or complete bounded distributive lattice on which the relevant selection principle holds while Player I has a winning strategy in the corresponding game. Theorem 6.5 proves the lower bounds $\operatorname{cov}\leq\epsilon^{\mathrm{pos}}_1$ and $$\mathfrak d\leq\epsilon^{\mathrm{pos}}_{\mathrm{fin}}$$, while the constructions provide upper bounds by the continuum.

It remains unresolved whether these lower bounds are exact in ZFC, whether restricting from posets to complete bounded distributive lattices changes the minima, and how these invariants relate to established cardinal characteristics of the continuum.