YES

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

Consider the left-linear TRS R:

  f(i(x),g(a())) -> f(j(x,x),g(b()))
  b() -> a()
  i(x) -> j(x,x)

Let C be the following subset of R:

  (empty)

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

  phi(f(i(x),g(a())) -> f(j(x,x),g(b()))) = 2
  phi(b() -> a()) = 1
  phi(i(x) -> j(x,x)) = 2

  psi(f(i(x),g(a())) -> f(j(x,x),g(b()))) = 2
  psi(b() -> a()) = 1
  psi(i(x) -> j(x,x)) = 2


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.