YES

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
f(x) -> g(f(x))
h(x) -> p(h(x))
f(x) -> h(f(x))
g(x) -> p(p(h(x)))

CP(R):
g(f(___y325)) -> h(f(___y325))
h(f(___y337)) -> g(f(___y337))

critical-pair-closing system C:
h(x) -> p(h(x))
g(x) -> p(p(h(x)))

CR(C): see below.

---- CR(R)
---- R : left-linear, C : CR and CP-closing, and Cdup/R : SN
R:
h(x) -> p(h(x))
g(x) -> p(p(h(x)))

CP(R):


critical-pair-closing system C:


CR(C): see below.

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