Unbounded envelope heights for finite subsets

Let $\str H$ and $\str G$ be the structures in the source, let $\varphi\colon\str H\to\str G$ be an embedding, and let an envelope of a finite subset have the source's valuation-tree meaning. Assume that for every $n\in\omega$ there is $a\in\omega$ such that the number of relations of arity $a$ is at least $n$. Unbounded envelope-height conjecture. For every embedding $\varphi\colon\str H\to\str G$, there is some $k\in\omega$ such that for every $R\in\omega$ there is a set $S\subseteq\varphi[\str H]$ with $|S|=k$ whose every envelope has height at least $R$. This strengthens the claim that no single embedding is uniformly $k$-enveloping.