These two properties are equivalent for:

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

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

These two properties are equivalent for:

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

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