YES

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

CP(R):


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

CR(C): see below.

---- CR(R)
---- R : linear and strongly closed [Huet, 1980]
R:
h(c(),c()) -> h(b(),f(b()))
h(c(),b()) -> f(a())
CP(R):