Let $X$ be a separable Banach space and $X^*$ be its dual, let $\{x_i\}$ be a sequence in $X$ with dense linear span and such that there exists a sequence $\{x_i^*\}$ in $X^*$ satisfying $x_i^*(x_j)=\delta_{i,j}$ (Kronecker delta). It is clear that $\{x_i^*\}$ is uniquely determined. Let us call $\{x_i\}$ a Markushevich–Auerbach basis if the linear span of $\{x_i^*\}$ is weak* dense in $X^*$ and $1=\|x_i\|_X=\|x_i^*\|_{X^*}$ for all $i$.

Question: Is the problem about the existence of a Markushevich–Auerbach basis in an arbitrary infinite-dimensional separable Banach space still open?

I am asking this question since I noticed that some recent papers (see Theorem 3.3 in http://arxiv.org/pdf/1604.03547.pdf) claim a solution of this problem, but I do not understand the argument.