YES

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

CP(R):
h(a(),h(f(b()),c())) -> c()

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

CR(C): see below.

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