NO
(ignored inputs)COMMENT from the collection of \cite{Aoto2013}
Rewrite Rules:
   [ a -> f(a),
     a -> g(h(a)),
     f(?x) -> h(g(?x)),
     h(?x) -> f(g(?x)) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ f(a) = g(h(a)) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
unknown Knuth & Bendix
Linear
unknown Development Closed
unknown Strongly Closed
unknown Weakly-Non-Overlapping & Non-Collapsing & Shallow
unknown Upside-Parallel-Closed/Outside-Closed
(inner) Parallel CPs: (not computed)
unknown Toyama (Parallel CPs)
Simultaneous CPs:
   [ g(h(a)) = f(a),
     f(a) = g(h(a)) ]
unknown Okui (Simultaneous CPs)
unknown Strongly Depth-Preserving & Root-E-Closed/Non-E-Overlapping
unknown Strongly Weight-Preserving & Root-E-Closed/Non-E-Overlapping
check Locally Decreasing Diagrams by Rule Labelling...
Critical Pair <g(h(a)), f(a)> by Rules <1, 0> preceded by []
unknown Diagram Decreasing
check Non-Confluence...
obtain 14 rules by 3 steps unfolding
strenghten f(f(a)) and f(a)
strenghten g(h(a)) and f(a)
strenghten h(g(a)) and f(a)
obtain 100 candidates for checking non-joinability
check by TCAP-Approximation (failure)
check by Root-Approximation (success)
Witness for Non-Confluence: <g(h(a)), f(a)>
Direct Methods: not CR

Final result: not CR
246.trs: Success(not CR)
(102 msec.)