5 problems

Hausdorff-measure lower-bound conjecture. There exists a constant $C>0$, depending on $n$, $k$, $hR$, and an upper bound for $\mathscr{H}^k(E)$, such that

Fixed-perimeter nonattainment conjecture. The infimum is not attained, and a minimizing sequence is given by any sequence of rectangles as in the shrinking-domain construction.

Fixed-diameter existence conjecture. The minimization problem

Fixed-perimeter nonattainment conjecture. The minimization problem

Eigenfunction regularity conjecture. Eigenfunctions belong to $C^\infty(\Omega) \cap C(\overline{\Omega})$.