Under what conditions does confluence of a normal semi-equational conditional term rewriting system imply confluence of the associated oriented system?

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

For a normal conditional term-rewriting system $R = \{ s \rightarrow^! t \Rightarrow l \rightarrow r \}$, where $t$ must be a ground normal from of $s$, we can consider the corresponding semi-equational conditional rewrite system $R_{se} = \{ s \leftrightarrow^* t \Rightarrow l \rightarrow r \}$. Under what conditions does confluence of $R_{se}$ imply confluence of $R$? In general, this is not the case, as can be seen from the following non-confluent system $R$ (due to Aart Middeldorp):

$$\begin{aligned} a \rightarrow b\\ a \rightarrow c\\ b \rightarrow^! c \Rightarrow b \rightarrow c \end{aligned}$$

Recorded progress.

Solutions have been provided by [Toyama, 1991]. They show that confluence of $R$ follows from confluence of $R_{se}$ if any of the two following conditions is satisfied: \begin{itemize}

1. $R_{se}$ is semi-decreasing

2. $R_{se}$ is level-confluent \end{itemize} See [Toyama, 1991] for definitions of these properties.