Discrete series conjecture for real reductive spherical spaces

Let $X_I$ be the spherical degeneration associated with a subset $I\subset S$, and let $i\mathfrak{a}_I^*$ denote the corresponding purely imaginary parameter space. Write $L^2(X_I,\nu)_{td}$ for the tempered discrete part at parameter $\nu$. Define $i\mathfrak{a}_{I,\mathrm{conj}}^*$ to be the set of $\nu\in i\mathfrak{a}_I^*$ such that either

$$L^2(X_I,\nu)_{td}=\{0\},$$

or a tempered discrete representation contained in $L^2(X_I,\nu)$ has an infinitesimal character whose some Harish-Chandra parameter has imaginary part equal to $\nu$.

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.

This is presented as a slightly weaker real version of the Discrete Series Conjecture of Sakellaridis and Venkatesh in the $p$-adic case. It is used in the paper's development of scattering theory and the Plancherel formula for real reductive spherical spaces; its resolution is not established in the supplied text.