Existence of a basic sequence with basis constant $1$ in a Banach space

Question

I am interested in a problem motivated by the following theorem. A proof (and the relevant definitions) can be found in many textbooks about Banach space theory, see for example Corollary 1.5.3 in Topics in Banach Space Theory by Albiac and Kalton.

Theorem. Let $X$ be an infinite-dimensional Banach space. Then for all $\epsilon > 0$, the space $X$ contains a basic sequence with basis constant at most $1 + \epsilon$.

Can we do any better than the above theorem? That is, does every infinite-dimensional Banach space contain a basic sequence with basis constant $1$ (monotone basic sequence)?

I think this is a natural question to ask, but I haven't made much progress so far and I also haven't found any references to this problem. Is there some easy counterexample or proof I'm missing?

Answer 1

This is an open problem. See problem 4 p. 220 in the book Banach space theory. The basis for linear and non-linear analysis. M. Fabian, P. Habala, P. Hajek, V. Montesinos, V. Zizler.