Classification conjecture for globally generated vector bundles with bounded first Chern class

Let $E$) be a globally generated vector bundle on $\mathbb{P}^n$, with $n\geq 2$ and $1\leq c_1(E)\leq n-1$, such that

$$H^i(E^*)=0\quad\text{for }i=0,1.$$

Write $P(B)$ for the bundle associated with a direct sum $B$ of line bundles as in the source. Classification conjecture. One of the following holds:

1. $E\simeq A\oplus P(B)$, where $A$ and $B$ are direct sums of line bundles on $\mathbb{P}^n$, with $c_1(A)+c_1(B)\leq n-1$, and $H^0(A^*)=H^0(B^*)=0$;
2. $n\geq 3$ and $E\simeq\Omega_{\mathbb{P}^n}(2)$;
3. $n\geq 4$ and $E\simeq\Omega_{\mathbb{P}^n}^{n-2}(n-1)$.

The claim proposes a classification of globally generated bundles when the first Chern class is bounded by the dimension. The supplied text presents it as a proposal motivated by the decreasing number of such bundles as $n$ increases; no resolution is given in the supplied material.