YES
Rewrite Rules:
   [ f(?x,?y) -> ?x,
     f(?x,?y) -> f(?x,g(?y)),
     g(?x) -> h(?x),
     F(g(?x),?x) -> F(?x,g(?x)),
     F(h(?x),?x) -> F(?x,h(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ F(h(?x_2),?x_2) = F(?x_2,g(?x_2)) ]
Outer CPs:
   [ ?x = f(?x,g(?y)) ]
unknown Terminating
not Left-Linear, not Right-Linear
unknown Gomi&Oyamaguchi&Ohta
check Non-Confluence...Unknown
Direct Methods: Can't judge

Try Persistent Decomposition for...
   [ f(?x,?y) -> ?x,
     f(?x,?y) -> f(?x,g(?y)),
     g(?x) -> h(?x),
     F(g(?x),?x) -> F(?x,g(?x)),
     F(h(?x),?x) -> F(?x,h(?x)) ]
Sort Assignment:
 F : 9*9=>18
 f : 21*9=>21
 g : 9=>9
 h : 9=>9
maximal types: {9,18}{9,21}
{9,18}
(ps)Rewrite Rules:
   [ g(?x) -> h(?x),
     F(g(?x),?x) -> F(?x,g(?x)),
     F(h(?x),?x) -> F(?x,h(?x)) ]
Apply Direct Methods...
Inner CPs:
   [ F(h(?x),?x) = F(?x,g(?x)) ]
Outer CPs:
   [  ]
Terminating, WCR: CR
{9,21}
(ps)Rewrite Rules:
   [ f(?x,?y) -> ?x,
     f(?x,?y) -> f(?x,g(?y)),
     g(?x) -> h(?x) ]
Apply Direct Methods...
Inner CPs:
   [  ]
Outer CPs:
   [ ?x = f(?x,g(?y)) ]
Overlay, check Innermost Termination...
unknown Innermost Terminating
Linear
Development Closed
Direct Methods: CR
Result by Persistent Decomposition: CR

Final result: CR
1.trs: YES (8 msec.)