Wqo-based decidability conjecture for the joint embedding property

Let $(\mathcal{F},\le)$ be a well-quasi-order (wqo), and suppose there are algorithms for basic problems related to $\le$, including deciding whether $X\le Y$ and, given $X$, finding all $Z$ such that $X\le Z$ and there is no $Y$ with $X\le Y\le Z$ other than $X$ and $Z$. For a finite set $S\subseteq\mathcal{F}$, write $\operatorname{Forb}_\le(S)$ for the objects containing no member of $S$ under $\le$. Wqo-based JEP conjecture. Under these assumptions, it should be decidable whether $\operatorname{Forb}_\le(S)$ has the joint embedding property (JEP). The conjecture is motivated by the possibility of extending the tree-automaton proof from wqo families of trees to more general wqo families, but the paper gives very low confidence in it and leaves the necessary additional assumptions unclear.