2 problems

Discrete series conjecture. For every $I\subset S$, the complement of $i\mathfrak{a}_{I,\mathrm{conj}}^*$ in $i\mathfrak{a}_I^*$ has Lebesgue measure zero.

Extended conjecture. If $\bar{L}_{i,n}>\bar{D}_{i,n}$ for $i=1,2$, then a sphere $\mathbb{S}^k$ is attached to $\mathcal{L}^{+\kappa_0}_{-\kappa_0}(\pmb{I},\pmb{Q})$, where…