YES

# Compositional parallel rule labeling (Shintani and Hirokawa 2022).

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  (empty)

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

  phi(h(f(),a(),a()) -> h(g(),a(),a())) = 4
  phi(h(g(),a(),a()) -> h(f(),a(),a())) = 8
  phi(a() -> a'()) = 7
  phi(h(x,a'(),y) -> h(x,y,y)) = 1
  phi(g() -> f()) = 1
  phi(f() -> g()) = 2

  psi(h(f(),a(),a()) -> h(g(),a(),a())) = 3
  psi(h(g(),a(),a()) -> h(f(),a(),a())) = 6
  psi(a() -> a'()) = 5
  psi(h(x,a'(),y) -> h(x,y,y)) = 2
  psi(g() -> f()) = 8
  psi(f() -> g()) = 6


Therefore, the confluence of R follows from that of C.

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

Consider the left-linear TRS R:

  (empty)

Let C be the following subset of R:

  (empty)

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.

# emptiness

The empty TRS is confluent.