Three-type existence claim for bounded biorthogonal systems

Let $\mathfrak X$ be an infinite-dimensional Banach space. A bounded biorthogonal system in $\mathfrak X$ is a sequence $\left\{x_n,x_n^*\right\}$ with $x_n\in\mathfrak X$ and $x_n^*\in\mathfrak X^*$ satisfying $x_n^*(x_m)=\delta_{nm}$ and with uniformly bounded norms. The three types considered are a $c_0$-biorthogonal system, a $wc_0^*$-stable biorthogonal system, and a DP-biorthogonal system.

Three-type existence claim. Every infinite-dimensional Banach space $\mathfrak X$ admits a bounded biorthogonal system $\left\{x_n,x_n^*\right\}$ that is one of the following three types: a $c_0$-biorthogonal system, a $wc_0^*$-stable biorthogonal system, or a DP-biorthogonal system.

The claim is presented as the result expected from combining earlier theorems on $c_0$, $wc_0^*$, and Dunford–Pettis-type biorthogonal systems. The supplied text does not establish its resolution, so its status remains open.