YES

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

Consider the left-linear TRS R:

  H(I(x)) -> K(J(x))
  J(x) -> K(J(x))
  I(x) -> I(J(x))
  J(x) -> J(K(J(x)))

Let C be the following subset of R:

  H(I(x)) -> K(J(x))
  J(x) -> K(J(x))
  I(x) -> I(J(x))
  J(x) -> J(K(J(x)))

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

  (empty)

All pairs in PCP(R) are joinable and PCPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Parallel rule labeling (Zankl et al. 2015).

Consider the left-linear TRS R:

  H(I(x)) -> K(J(x))
  J(x) -> K(J(x))
  I(x) -> I(J(x))
  J(x) -> J(K(J(x)))

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

  phi(H(I(x)) -> K(J(x))) = 2
  phi(J(x) -> K(J(x))) = 1
  phi(I(x) -> I(J(x))) = 3
  phi(J(x) -> J(K(J(x)))) = 1

  psi(H(I(x)) -> K(J(x))) = 2
  psi(J(x) -> K(J(x))) = 1
  psi(I(x) -> I(J(x))) = 3
  psi(J(x) -> J(K(J(x)))) = 1