YES

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

CP(R):


critical-pair-closing system C:
g(a(),y) -> y
g(h(x),y) -> g(x,h(y))

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
g(a(),y) -> y
g(h(x),y) -> g(x,h(y))
CP(R):