YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a() -> c()
c() -> a()
c() -> h(a(),c())

CP(R):
a() -> h(a(),c())
h(a(),c()) -> a()

critical-pair-closing system C:
a() -> c()
c() -> h(a(),c())

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
a() -> c()
c() -> h(a(),c())

CP(R):


critical-pair-closing system C:


CR(C): see below.

---- CR(R)
---- R is empty.