Fano schemes of sub-maximal elementary symmetric functions

Denote by $E_r$ the $r^{th}$ elementary symmetric polynomial in $\dim V$ variables for a vector space $V$ over an infinite field $\mathbb{k}$. We describe the rational points on the Fano scheme $F_{d-1}(Z(E_{\dim V-1}))$ of projective $(d-1)$-spaces contained in the zero locus of $E_{\dim V-1}$. Isolated points exist precisely for $\dim V=2d$, in which case they are in bijection with the $1\cdot 3\cdots (2d-1)$ pairings on a $2d$-element set. This, in particular, confirming a conjecture of Ambartsoumian, Auel and Jebelli to the effect that (over $\mathbb{R}$) all isolated points are recoverable via integral star transforms with appropriate symbols.