YES

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

Consider the left-linear TRS R:

  g(a()) -> f(g(a()))
  g(b()) -> c(a())
  a() -> b()
  f(x) -> h(x)
  h(x) -> c(b())

Let C be the following subset of R:

  g(a()) -> f(g(a()))
  g(b()) -> c(a())
  a() -> b()
  f(x) -> h(x)
  h(x) -> c(b())

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:

  g(a()) -> f(g(a()))
  g(b()) -> c(a())
  a() -> b()
  f(x) -> h(x)
  h(x) -> c(b())

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

  phi(g(a()) -> f(g(a()))) = 3
  phi(g(b()) -> c(a())) = 1
  phi(a() -> b()) = 2
  phi(f(x) -> h(x)) = 1
  phi(h(x) -> c(b())) = 1

  psi(g(a()) -> f(g(a()))) = 4
  psi(g(b()) -> c(a())) = 1
  psi(a() -> b()) = 3
  psi(f(x) -> h(x)) = 1
  psi(h(x) -> c(b())) = 1