A remark similar to Hailong Dao's comment under his answer:

Let $E$ be a vector bundle on $\mathbb{P}^n$. A cohomological criterion (Horrocks' criterion) states that $E$ splits if and only if $H^i(\mathbb{P}^n, E(t))=0$ for $0 < i < n$ and all $t$.

There is a little less well known criterion, due to Evans and Griffiths, which says that we only need to check the vanishing of $H^i(\mathbb{P}^n, E(t))$ for $0 < i < \min(n, rank(E))$ and all $t$.

In particular, in the rank two case, the whole conjecture boils down to the simple claim that $H^1(\mathbb{P}^n, E) = 0$. Since $E$ is trivial on each "standard open" $U_i$, we can describe cohomology classes in this $H^1$ group using explicit Cech cocycles in this covering.

In summary, it is surprising how little we know about such a simple situation!