YES

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

Consider the left-linear TRS R:

  H(H(x)) -> K(x)
  H(K(x)) -> K(H(x))

Let C be the following subset of R:

  H(H(x)) -> K(x)
  H(K(x)) -> K(H(x))

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

  phi(H(H(x)) -> K(x)) = 0
  phi(H(K(x)) -> K(H(x))) = 0

  psi(H(H(x)) -> K(x)) = 0
  psi(H(K(x)) -> K(H(x))) = 0


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:

  H(H(x)) -> K(x)
  H(K(x)) -> K(H(x))

Let C be the following subset of R:

  (empty)

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

  H(H(H(x1_1))) -> H(K(x1_1))
  H(H(H(x1_1))) -> K(H(x1_1))
  H(H(K(x1_1))) -> H(K(H(x1_1)))
  H(H(K(x1_1))) -> K(K(x1_1))

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.