These two properties are equivalent for:

 - Dual spaces (R. E. Huff, P. D. Morris, [*Proc. Amer. Math. Soc.* **49** (1975), 104-108][1], C. Stegall, [*Trans. Amer. Math. Soc.* **206** (1975), 213-223][2], C. Stegall, [*Trans. Amer. Math. Soc.* **264** (1981), 507-519][3]);
 - Banach spaces isomorphic to their squares (W. Schachermayer, [*Studia Math.* **81** (1985), 329-339][4]).
 - certain spaces with a shrinking basis (G. López, J. F. Mena, [Studia Math. 118 (1996), 11-17][5]).


  [1]: https://www.ams.org/journals/tran/1987-303-02/S0002-9947-1987-0902791-8/S0002-9947-1987-0902791-8.pdf
  [2]: https://www.ams.org/journals/tran/1975-206-00/S0002-9947-1975-0374381-1/S0002-9947-1975-0374381-1.pdf
  [3]: https://www.ams.org/journals/tran/1981-264-02/S0002-9947-1981-0603779-1/S0002-9947-1981-0603779-1.pdf
  [4]: https://www.impan.pl/en/publishing-house/journals-and-series/studia-mathematica/all/81/3/104710/for-a-banach-space-isomorphic-to-its-square-the-radon-nikodym-property-and-the-krein-milman-property-are-equivalent
  [5]: https://www.infona.pl/resource/bwmeta1.element.bwnjournal-article-smv118i1p11bwm#

I do not think any significant progress has been obtained since.