The Alternate Thomassé conjecture on sibling numbers of relations

Let $R=(V,E)$ be a relation of arbitrary cardinality, where $V$ is a non-empty domain and $E$ is an $n$-ary relation on $V$ for some positive integer $n$. The sibling number $Sib(R)$ is the number of isomorphism classes of relations equimorphic to $R$, where two relations are equimorphic when each embeds in the other.

The Alternate Thomassé conjecture. For every relation $R$ of arbitrary cardinality,

$$Sib(R)=1\quad\text{or}\quad\infty.$$

This is the unrestricted-cardinality alternative to Thomassé's conjecture and is the version addressed for countable $NE$-free posets in the paper. The supplied text gives no evidence that the conjecture has been resolved in general.