Schmitz's path characterization of labelled well-quasi-orderings

Let $\mathcal{C}$ be a class of finite relational structures. A formula $\varphi(x,y)$ defines a path of length $n$ in a structure if there exist $n$ distinct elements $a_0,\ldots,a_{n-1}$ such that $\varphi(a_i,a_j)$ holds if and only if $|i-j|=1$. A class labelled by a well-quasi-ordered set $(Y,\leq)$ is well-quasi-ordered under the labelled-induced-substructure relation when every such labelled class has this property.

Schmitz's conjecture. The following are equivalent: (i) the class of structures in $\mathcal{C}$ labelled with any well-quasi-ordered set $(Y,\leq)$ is itself well-quasi-ordered under the labelled-induced-substructure relation; (ii) for every existential formula $\varphi(x,y)$, there exists $N_\varphi\in\mathbb{N}$ such that $\varphi$ does not define paths of length greater than $N_\varphi$ in structures of $\mathcal{C}$.

This generalizes Pouzet's conjecture. A negative answer is known for finite non-relational structures, while the conjecture remains open for finite relational structures.