Existence of uniform posets that are not monoid posets

A uniform poset is a poset such that, for every pair of elements $x,y$, the principal upsets ${\uparrow\hspace{-1.5pt}} x$ and ${\uparrow\hspace{-1.5pt}} y$ are order-isomorphic.

Uniform-poset nonexistence conjecture. There are uniform posets that are not monoid posets.

Uniformity generalizes the auto-equivalent setting, and the supplied context says that auto-equivalent posets are uniform. The existence of a uniform poset outside the class of monoid posets is presented as a suspicion and remains open in the supplied text.