Longest Sublist with Equal Occurrences
Jelly, 8 7 bytes
ẆĠẈEƊƇṪ

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How it works
ẆĠẈEƊƇṪ - Main link. Takes L on the left
Ẇ       - All non-empty contiguous sublists of L, longest last
    ƊƇ  - Keep those sublists S for which the following is true:
 Ġ      -   Group the indices of S by the elements
  Ẉ     -   Length of each
   E    -   All equal?
      Ṫ - Get the last (i.e. the longest) element