Optimal lower bounds for alpha-undominated sets

Let an election have committee size parameter $k$ and threshold $\alpha$, and let an $\alpha$-undominated set mean a committee satisfying the paper's definition of $\alpha$-undomination.

Lower-bound optimality conjecture. If $\alpha \geq \frac{2}{k+1}$, then an $\alpha$-undominated set of size $k$ always exists.

The conjecture would close the gap between the paper's upper and lower bounds and would imply that every election has Condorcet dimension at most $3$. The case $k=2$ is highlighted as particularly interesting; the parser supplies no resolution status, and the paper notes reasons for skepticism based on several constructions of lower bounds.