YES

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

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

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

CR(C): see below.

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