YES

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

  f(f(f(x))) -> a()
  f(f(a())) -> a()
  f(a()) -> a()
  f(f(g(g(x)))) -> f(a())
  g(f(a())) -> a()
  g(a()) -> a()

Let C be the following subset of R:

  f(f(g(g(x)))) -> f(a())
  g(a()) -> a()
  f(a()) -> a()
  f(f(f(x))) -> a()
  f(f(a())) -> a()

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

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

Consider the left-linear TRS R:

  f(f(g(g(x)))) -> f(a())
  g(a()) -> a()
  f(a()) -> a()
  f(f(f(x))) -> a()
  f(f(a())) -> a()

Let C be the following subset of R:

  (empty)

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

  f(f(g(g(a())))) -> f(f(g(a())))
  f(f(g(g(a())))) -> f(a())
  f(f(f(g(g(x1_1))))) -> f(f(a()))
  f(f(f(g(g(x1_1))))) -> a()
  f(f(f(f(x1_1)))) -> f(a())
  f(f(f(f(x1_1)))) -> a()
  f(f(f(a()))) -> f(a())
  f(f(f(a()))) -> a()
  f(f(f(f(g(g(x2_1)))))) -> f(f(f(a())))
  f(f(f(f(g(g(x2_1)))))) -> a()
  f(f(f(a()))) -> f(f(a()))
  f(f(f(f(f(x2_1))))) -> f(f(a()))
  f(f(f(f(f(x2_1))))) -> a()
  f(f(f(f(a())))) -> f(f(a()))
  f(f(f(f(a())))) -> a()
  f(f(a())) -> f(a())
  f(f(a())) -> a()

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.