Lifting conjecture for instanton sheaves on projective spaces

Let $E$ be a rank $r\geq 2n$ locally-free instanton sheaf of trivial splitting type on $\mathbb{P}^{2n}$ and charge $c$. Lifting conjecture. There is a rank $r$ locally-free instanton sheaf $\tilde{E}$ on $\mathbb{P}^{2n+1}$ of charge $c$ and a hyperplane $\wp\subset\mathbb{P}^{2n+1}$ such that

$$\tilde{E}|_{\wp}\simeq E.$$

This would extend the known lifting result from $\mathbb{P}^{2}$ to higher-dimensional projective spaces. The statement is presented as a proposed generalization, and no resolution is supplied here.