Is the word problem for all proper combinators of order smaller than 3 decidable?

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

The order of a proper combinator is the number of variables on the left hand side of its defining equation. For instance, the $K$ combinator has order 2. Is the word problem for all proper combinators of order smaller than 3 decidable? See [Statman, 2000] for related results.

A related question is the word problem for the $S$-combinator (of order 3), see RTALooP entry Scomb-word.