Giambelli compatibility implies determinantal structure

Let $E$ be the underlying space, let $\mathbb{P}$ be a point process on $E$, and let $\rho^R$ be an $L^1$-specialization. The pair $(\mathbb{P},\rho^R)$ is Giambelli compatible when it satisfies the Giambelli-compatibility condition described in the source. Giambelli compatibility conjecture. If $(\mathbb{P},\rho^R)$ is Giambelli compatible, then $\mathbb{P}$ is a determinantal point process. This is a converse to deriving Giambelli compatibility from determinantal structure and concerns the relation between symmetric-function identities and correlation functions of point processes; the supplied text does not establish whether the assertion is proved or remains open.