YES

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

Consider the left-linear TRS R:

  f(x1,g(x2)) -> f(x1,g(x1))
  f(g(y1),y2) -> f(g(y1),g(y1))
  g(a()) -> g(b())
  b() -> a()

Let C be the following subset of R:

  f(x1,g(x2)) -> f(x1,g(x1))
  f(g(y1),y2) -> f(g(y1),g(y1))
  b() -> 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:

  f(x1,g(x2)) -> f(x1,g(x1))
  f(g(y1),y2) -> f(g(y1),g(y1))
  b() -> a()

Let C be the following subset of R:

  f(g(y1),y2) -> f(g(y1),g(y1))

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:

  f(g(y1),y2) -> f(g(y1),g(y1))

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.