Generalized Cayley-Bacharach conjecture for quadrics

Let $\Gamma$ be a complete intersection of $n$ quadrics in $\mathbb{P}^n$. Let $X\subset\mathbb{P}^n$ be a hypersurface of degree $d$ containing a subscheme $\Omega\subset\Gamma$ whose degree is strictly greater than $2^n-2^{n-d}$. Generalized Cayley-Bacharach conjecture for quadrics. Then $X$ must contain $\Gamma$. This geometric conjecture is presented as a motivation for the quadratic EGH conjecture and is implied by it. The source does not state a resolution, so its status is open.