Model completion for trace-monoid derivations

Let $\Lambda$ be a partial commutative monoid generated by a tuple $\bar\delta$; equivalently, let $\Lambda$ be a trace monoid (Cartier–Foata monoid). Let $T^\Lambda$ denote the corresponding theory of derivation actions. Trace-monoid model-completion conjecture. $T^\Lambda$ has a model completion. The claim concerns extending model-completion results from the generic derivation setting to partial commutative, or trace-monoid, derivations; it is presented as an open problem and remains unresolved in the source.