YES

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

Consider the left-linear TRS R:

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

Let C be the following subset of R:

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

# parallel critical pair closing system (Shintani and Hirokawa 2022)

Consider the left-linear TRS R:

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

Let C be the following subset of R:

  g() -> f()
  a() -> a'()

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

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

Consider the left-linear TRS R:

  g() -> f()
  a() -> a'()

Let C be the following subset of R:

  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.

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

Consider the left-linear TRS R:

  a() -> a'()

Let C be the following subset of R:

  (empty)

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.

# emptiness

The empty TRS is confluent.