YES

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

Consider the left-linear TRS R:

  f(a(),a()) -> b()
  a() -> a'()
  f(a'(),x) -> f(x,x)
  f(x,a'()) -> f(x,x)
  f(a'(),a'()) -> b()
  b() -> f(a'(),a'())

Let C be the following subset of R:

  a() -> a'()
  b() -> 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)

Consider the left-linear TRS R:

  a() -> a'()
  b() -> f(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 follows from that of C.

# emptiness

The empty TRS is confluent.