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?