Nowadays, we know that there exist Banach spaces without unconditional basic sequences. Do we know if something a bit milder holds? Namely, is that true each non-reflexive Banach space contains a normalised basic sequence $(x_n)_{n=1}^\infty$ such that for each $(t_n)_{n=1}^\infty\in c_0$ the series

$$\sum_{n=1}^\infty t_n x_n$$

is convergent in this space? Has anyone studied this sort of basic sequences?