YES

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

Consider the left-linear TRS R:

  foo(0(x)) -> 0(s(p(p(p(s(s(s(p(s(x))))))))))
  foo(s(x)) -> p(s(p(p(p(s(s(p(s(s(p(s(foo(p(p(s(s(p(s(bar(p(p(s(s(p(s(x))))))))))))))))))))))))))
  bar(0(x)) -> 0(p(s(s(s(x)))))
  bar(s(x)) -> p(s(p(p(s(s(foo(s(p(p(s(s(x))))))))))))
  p(p(s(x))) -> p(x)
  p(s(x)) -> x
  p(0(x)) -> 0(s(s(s(s(x)))))

Let C be the following subset of R:

  (empty)

The critical pair system CPS(R,C) is:

  p(p(s(y0))) -> p(y0)

The TRS R is locally confluent and CPS(R,C)/R is terminating.
Therefore, the confluence of R follows from that of C.

# Emptiness.

The empty TRS is confluent.