YES

# parallel critical pair closing system (Shintani and Hirokawa 2022, Section 8 in LMCS 2023)

Consider the left-linear TRS R:

  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  hd(:(x,y)) -> x
  tl(:(x,y)) -> y
  inc(tl(nats())) -> tl(inc(nats()))

Let C be the following subset of R:

  inc(tl(nats())) -> tl(inc(nats()))
  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

The TRS R is left-linear and all parallel critical pairs are joinable by C.
Therefore, the confluence of R is equivalent to that of C.

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

Consider the left-linear TRS R:

  inc(tl(nats())) -> tl(inc(nats()))
  nats() -> :(0(),inc(nats()))
  inc(:(x,y)) -> :(s(x),inc(y))
  tl(:(x,y)) -> y

Let C be the following subset of R:

  (empty)

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

  inc(tl(nats())) -> inc(tl(:(0(),inc(nats()))))
  inc(tl(nats())) -> tl(inc(nats()))

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.