SmallGroup(128,227) and SmallGroup(128,731) seem to be counterexamples.

SmallGroup(128,227) and SmallGroup(128,731)) are counterexamples.

gap> S:=List([227,731],n->SmallGroup(128,n));;
gap> for g in S do L:=List(ConjugacyClasses(g),c->Size(c));; Sort(L);; Print(L); od;
[ 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8 ]
[ 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8 ]
gap> for g in S do Print(CharacterDegrees(g)); od;
[ [ 1, 16 ], [ 2, 12 ], [ 4, 4 ] ]
[ [ 1, 8 ], [ 2, 22 ], [ 4, 2 ] ]