Infinite big Ramsey degrees with infinitely many relations of one arity

Let $L$ be a relational language with finitely many unary relations and infinitely many relations of some arity $a\geq 2$. Let $\str H$ be a countable unrestricted $L$-structure realising all relations from $L$. Infinite big Ramsey degree conjecture. There is a finite $L$-structure $\str A$ whose big Ramsey degree in $\str H$ is infinite. Moreover, the number of vertices of $\str A$ depends only on $a$. The conjecture proposes that the finite big Ramsey degree theorem is sharp once infinitely many relations occur in a fixed arity at least two.