Not much is known about vector bundles on $\mathbb{P}^2$ but I wonder if the following is a tractable question:

If $E,E'$ are non-isomorphic vector bundles on $\mathbb{P}^2$, then is there always a smooth curve $C \subset \mathbb{P}^2$ such that $E|_C$ and $E'|_C$ are still non isomorphic?

A related and perhaps easier question: Can a vector bundle restrict to the trivial bundles on every curve without itself being trivial on $\mathbb{P}^2$?