Which conditional rewrite systems are subcommutative?

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

Parallel rewriting with orthogonal term-rewriting systems is subcommutative'' (a strong'' form of confluence). Under which interesting syntactic restrictions do conditional rewrite systems enjoy the same property? It is known that orthogonal normal'' conditional rewriting systems (with conditions $u \rightarrow^! v$, where $v$ is a ground normal form) are confluent, while standard'' (join) ones are not [Jouannaud, 1991].