YES

Problem:
 0(0(*(*(x1)))) -> *(*(1(1(x1))))
 1(1(*(*(x1)))) -> 0(0(#(#(x1))))
 #(#(0(0(x1)))) -> 0(0(#(#(x1))))
 #(#(1(1(x1)))) -> 1(1(#(#(x1))))
 #(#($($(x1)))) -> *(*($($(x1))))
 #(#(#(#(x1)))) -> #(#(x1))
 #(#(*(*(x1)))) -> *(*(x1))

Proof:
 Arctic Interpretation Processor:
  dimension: 1
  interpretation:
   [$](x0) = 8x0,
   
   [#](x0) = 1x0,
   
   [1](x0) = x0,
   
   [0](x0) = x0,
   
   [*](x0) = 1x0
  orientation:
   0(0(*(*(x1)))) = 2x1 >= 2x1 = *(*(1(1(x1))))
   
   1(1(*(*(x1)))) = 2x1 >= 2x1 = 0(0(#(#(x1))))
   
   #(#(0(0(x1)))) = 2x1 >= 2x1 = 0(0(#(#(x1))))
   
   #(#(1(1(x1)))) = 2x1 >= 2x1 = 1(1(#(#(x1))))
   
   #(#($($(x1)))) = 18x1 >= 18x1 = *(*($($(x1))))
   
   #(#(#(#(x1)))) = 4x1 >= 2x1 = #(#(x1))
   
   #(#(*(*(x1)))) = 4x1 >= 2x1 = *(*(x1))
  problem:
   0(0(*(*(x1)))) -> *(*(1(1(x1))))
   1(1(*(*(x1)))) -> 0(0(#(#(x1))))
   #(#(0(0(x1)))) -> 0(0(#(#(x1))))
   #(#(1(1(x1)))) -> 1(1(#(#(x1))))
   #(#($($(x1)))) -> *(*($($(x1))))
  KBO Processor:
   weight function:
    w0 = 1
    w($) = w(#) = w(1) = w(0) = w(*) = 1
   precedence:
    # > 1 > 0 > $ ~ *
   problem:
    
   Qed