Pemantle's h-NLC and Rayleigh equivalence conjecture

Let CNA and CNA+ denote the corresponding conditional negative-association properties, and let h-NLC and h-NLC+ denote the hereditary negative-lattice-condition properties; Rayleigh is the Rayleigh property. Pemantle's equivalence conjecture. The implications $\mathrm{CNA}\Rightarrow\mathrm{h\text{-}NLC}$ and $\mathrm{CNA+}\Rightarrow\mathrm{Rayleigh}/\mathrm{h\text{-}NLC+}$ are equivalences, namely

$$\mathrm{h\text{-}NLC}\Rightarrow\mathrm{CNA}\quad\text{and}\quad \mathrm{Rayleigh}/\mathrm{h\text{-}NLC+}\Rightarrow\mathrm{CNA+}.$$

The source presents these as open equivalence questions; no resolution is supplied in the given text.