Is satisfiability of lpo or rpo ordering constraints decidable in case of non-total precedences?

The source page states the original problem together with its recorded qualifications and progress updates as follows.

The existential fragment of the first-order theory of the recursive path ordering'' (with multiset and lexicographic status'') is decidable when the precedence on function symbols is total [Jouannaud, 1991], but is undecidable for arbitrary formulas. Is the existential fragment decidable for partial precedences?

Recorded progress.

The $\Sigma_4$ ($\exists^*\forall^*\exists^*\forall^*$) fragment is undecidable, in general [Jouannaud, 1991]. The positive existential fragment for the empty precedence (that is, for homeomorphic tree embedding) is decidable [Jouannaud, 1991]. One might also ask whether the first-order theory of total recursive path orderings is decidable. Related results include the following: The existential fragment of the subterm ordering is decidable, but its $\Sigma_2$ ($\exists^*\forall^*$) fragment is not [Jouannaud, 1991]. The first-order theory of encompassment (the instance-of-subterm relation) is decidable [Jouannaud, 1991]. The satisfiability problem for the existential fragment in the total case is NP-complete [Jouannaud, 1991].

Though the first-order theory of encompassment is decidable [Jouannaud, 1991], the first-order ($\Sigma_2$) theory of the recursive (lexicographic status) path ordering, assuming certain simple conditions on the precedence, is not [Jouannaud, 1991].