Is the word problem for the S-combinator decidable?

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

The word problem for the $S$-combinator is: Given two ground terms build only of the constant $S$ in combinatory logic (that is with an application operator written as juxtaposition, and parantheses), are they convertible in the system consisting only of the definition of the $S$-combinator

$$S x y z \rightarrow (x z) (y z)$$

Is the word-problem for the $S$-combinator decidable? See also [Barendregt, 1975] and [Barendregt, 1975] for more background.

A related problem is the word problem for proper combinators of order smaller than 3 ($S$ is of order 3), see RTALooP entry smallorder-combinators-word.