YES

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

Consider the left-linear TRS R:

  f(g(x)) -> h(x,x)
  g(a()) -> b()
  f(x) -> h(x,x)
  b() -> a()
  h(x,y) -> h(g(x),g(y))
  g(x) -> x
  a() -> b()

Let C be the following subset of R:

  f(g(x)) -> h(x,x)
  g(a()) -> b()
  f(x) -> h(x,x)
  b() -> a()
  h(x,y) -> h(g(x),g(y))
  g(x) -> x
  a() -> 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:

  f(g(x)) -> h(x,x)
  g(a()) -> b()
  f(x) -> h(x,x)
  b() -> a()
  h(x,y) -> h(g(x),g(y))
  g(x) -> x
  a() -> b()

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

  phi(f(g(x)) -> h(x,x)) = 9
  phi(g(a()) -> b()) = 6
  phi(f(x) -> h(x,x)) = 4
  phi(b() -> a()) = 3
  phi(h(x,y) -> h(g(x),g(y))) = 1
  phi(g(x) -> x) = 8
  phi(a() -> b()) = 2

  psi(f(g(x)) -> h(x,x)) = 4
  psi(g(a()) -> b()) = 8
  psi(f(x) -> h(x,x)) = 5
  psi(b() -> a()) = 1
  psi(h(x,y) -> h(g(x),g(y))) = 1
  psi(g(x) -> x) = 3
  psi(a() -> b()) = 7