YES

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

CP(R):


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

CR(C): see below.

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