Local character of finite and partial function categories

Let $\Xi$ be the universe of categories enriched in pointed DCPOs and Scott-continuous functors between them, respecting finite coproducts and initial objects. Let $\Theta$ be the universe of symmetric monoidal categories enriched in pointed DCPOs and Scott-continuous monoidal functors between them, respecting finite coproducts and initial objects. Let $\operatorname{fpFun}$ be the subcategory of finite prime ordinals with total functions between them, and let $\operatorname{pFun}$ be the subcategory of finite prime ordinals and infinite initial ordinals with total functions between them. Local-character conjecture. The category $\operatorname{fpFun}$ has local character with respect to $(\Xi,\Theta)$, and the category $\operatorname{pFun}$ has local character with respect to $(\Xi,\Theta)$. This is presented as a tentative result in the enriched categorical setting; the source leaves further investigation to future work, and no resolution is given.