Answer by Nick Switala for Independence of embedding for higher sheaf cohomology of local cohomology on projective space

It is false! As mentioned in the edit, a positive answer to this question would imply the Lyubeznik numbers $\lambda_{i,j}$ of a projective scheme are independent of the defining projective embedding for $i \geq 2$. The recent preprint

https://arxiv.org/abs/1803.07448

by T. Reichelt, M. Saito, and U. Walther, gives counterexamples (even equidimensional ones) in characteristic zero.