Hartshorne's unstable rank-two bundle conjecture

Let $n$ be an integer with $n\geq n_0'$, where $n_0'\geq 5$, and let $E$ be a rank $2$ vector bundle on $\mathbb{P}^n$. The bundle $E$ is unstable in the sense used in the source, and a vector bundle is decomposable if it is a direct sum of line bundles.

Hartshorne's unstable rank-two bundle conjecture. There exists $n_0'\geq 5$ such that any unstable rank $2$ vector bundle $E$ on~$\mathbb{P}^n$, with $n\geq n_0'$, is decomposable.

This is presented as an interesting and very plausible unstable variant of Hartshorne's conjecture. It is studied in the cited literature, but the proofs there have serious gaps, so the conjecture remains open.