Is an ultra-net a subnet of all it's subnets?

Question

Let $(x_d)_{d\in D}$ be a net on a set $X$ and the set $$\{\{x_d\mid d\ge p\}\mid p\in D \}$$ be a base for an ultrafilter on $X$.

Let $(x_{d'})_{d'\in D'}$ be a subnet of $(x_d)_{d\in D}$. Is $(x_d)_{d\in D}$ a subnet of $(x_{d'})_{d'\in D'}$?

Answer 1

An ultranet (or universal net) has no proper subnets, so the answer is yes; see Lemma $13.3$ of this PDF by Saitulaa Naranong. (Note that the implication in the displayed line of Definition $10.2$ is backwards: $\Phi$ and $\Psi$ should be interchanged.)