YES

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

Consider the left-linear TRS R:

  b(w(x)) -> w(w(w(b(x))))
  w(b(x)) -> b(x)
  b(b(x)) -> w(w(w(w(x))))
  w(w(x)) -> w(x)

Let C be the following subset of R:

  b(w(x)) -> w(w(w(b(x))))
  w(b(x)) -> b(x)
  b(b(x)) -> w(w(w(w(x))))
  w(w(x)) -> w(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:

  b(w(x)) -> w(w(w(b(x))))
  w(b(x)) -> b(x)
  b(b(x)) -> w(w(w(w(x))))
  w(w(x)) -> w(x)

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

  phi(b(w(x)) -> w(w(w(b(x))))) = 3
  phi(w(b(x)) -> b(x)) = 4
  phi(b(b(x)) -> w(w(w(w(x))))) = 5
  phi(w(w(x)) -> w(x)) = 3

  psi(b(w(x)) -> w(w(w(b(x))))) = 6
  psi(w(b(x)) -> b(x)) = 1
  psi(b(b(x)) -> w(w(w(w(x))))) = 7
  psi(w(w(x)) -> w(x)) = 2