The generalized-quasiorder characterization conjecture for partial orders

Let $A$ be a set and let $\rho$ be a partial order on $A$ with least element $0$ and greatest element $1$. Let $\operatorname{Pol}\rho$ be the clone of operations preserving $\rho$, let $[\rho]_{(\land,=)}$ denote the relations obtained from $\rho$ using conjunction and equality, and let $[\rho]_{(\exists,\land,=)}$ denote those obtained using existential quantification, conjunction and equality. Write $\operatorname{gQuord}(A)$ for the generalized quasiorders on $A$. The generalized-quasiorder characterization conjecture.

$$\operatorname{gQuord}\operatorname{Pol}\rho=[\rho]_{(\land,=)}.$$

Equivalently,

$$[\rho]_{(\exists,\land,=)}\cap\operatorname{gQuord}(A)=[\rho]_{(\land,=)}.$$

This is a modest version of the preceding conjectural equality, restricted to partial orders with least and greatest elements; lattice orders are already covered by the paper's theorem.