YES

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

Consider the left-linear TRS R:

  f(a(),a(),b(),b()) -> f(c(),c(),c(),c())
  a() -> b()
  a() -> c()
  b() -> a()
  b() -> c()

Let C be the following subset of R:

  f(a(),a(),b(),b()) -> f(c(),c(),c(),c())
  a() -> b()
  a() -> c()
  b() -> a()
  b() -> c()

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

  (empty)

The TRS R is locally confluent 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:

  f(a(),a(),b(),b()) -> f(c(),c(),c(),c())
  a() -> b()
  a() -> c()
  b() -> a()
  b() -> c()

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

  phi(f(a(),a(),b(),b()) -> f(c(),c(),c(),c())) = 3
  phi(a() -> b()) = 1
  phi(a() -> c()) = 3
  phi(b() -> a()) = 2
  phi(b() -> c()) = 2

  psi(f(a(),a(),b(),b()) -> f(c(),c(),c(),c())) = 5
  psi(a() -> b()) = 4
  psi(a() -> c()) = 5
  psi(b() -> a()) = 3
  psi(b() -> c()) = 5