YES

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

CP(R):
p(___y3) -> p(___y3)
s(___y5) -> s(___y5)

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

CR(C): see below.

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