Is any ``strongly'' non-overlapping right-linear term-rewriting system confluent?

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

Is any strongly'' non-overlapping right-linear term-rewriting system confluent? (Strong'' in the sense that left-hand sides are non-overlapping even when the occurrences of variables have been renamed apart [Oyamaguchi, 1993].) On the one hand, strongly non-overlapping systems need not be confluent [Oyamaguchi, 1993]; on the other hand, strongly non-overlapping right-ground systems are [Oyamaguchi, 1993].

Recorded progress.

A partial positive solution is given in [Oyamaguchi, 1993], namely, any strongly non-overlapping right-linear term-rewriting system is confluent if it satisfies the condition that for any rewrite rule, no variables occurring more than once in the left-hand-side occur in the right-hand-side.