YES

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

Consider the left-linear TRS R:

  F(H(x),y) -> F(H(x),I(I(y)))
  F(x,G(y)) -> F(I(x),G(y))
  I(x) -> x

Let C be the following subset of R:

  F(H(x),y) -> F(H(x),I(I(y)))
  F(x,G(y)) -> F(I(x),G(y))
  I(x) -> 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:

  F(H(x),y) -> F(H(x),I(I(y)))
  F(x,G(y)) -> F(I(x),G(y))
  I(x) -> x

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

  phi(F(H(x),y) -> F(H(x),I(I(y)))) = 2
  phi(F(x,G(y)) -> F(I(x),G(y))) = 3
  phi(I(x) -> x) = 1

  psi(F(H(x),y) -> F(H(x),I(I(y)))) = 2
  psi(F(x,G(y)) -> F(I(x),G(y))) = 3
  psi(I(x) -> x) = 1