YES

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

CP(R):
q(___y3) -> r(___y3)
r(___y11) -> q(___y11)

critical-pair-closing system C:
p(x) -> q(x)
q(x) -> s(p(x))
r(x) -> s(p(x))
s(x) -> f(p(x))

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
p(x) -> q(x)
q(x) -> s(p(x))
r(x) -> s(p(x))
s(x) -> f(p(x))
CP(R):