## 1-product sets in finite decomposable sets in groups

A non-empty subset $$D$$ of a group is called decomposable if each element $$x\in D$$ can be written as the product $$x=yz$$ for some $$y,z\in D$$.

Problem. Let $$D$$ be a finite decomposable subset of a group. Is there a sequence $$x_1,\dots,x_n$$ of pairwise distinct points of $$D$$ such that $$x_1\cdots x_n=1$$?

Remark. This problem is a non-commutative version of this still open problem posed by Zaimi in 2010 on MO. So, maybe in the non-commutative case there is a counterexample?