YES

# Compositional parallel critical pair system (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

  W(B(x)) -> W(x)
  B(I(x)) -> J(x)
  W(I(x)) -> W(J(x))

Let C be the following subset of R:

  W(B(x)) -> W(x)
  B(I(x)) -> J(x)
  W(I(x)) -> W(J(x))

The parallel critical pair system PCPS(R,C) is:

  (empty)

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  W(B(x)) -> W(x)
  B(I(x)) -> J(x)
  W(I(x)) -> W(J(x))

All parallel critical peaks (except C's) are decreasing wrt rule labeling:

  phi(W(B(x)) -> W(x)) = 2
  phi(B(I(x)) -> J(x)) = 2
  phi(W(I(x)) -> W(J(x))) = 1

  psi(W(B(x)) -> W(x)) = 2
  psi(B(I(x)) -> J(x)) = 2
  psi(W(I(x)) -> W(J(x))) = 1