Let $X$ be a Stein manifold with analytic structure sheaf $\mathscr{O}_{X}$. Let $M$ be a coherent $\mathscr{O}_{X}$-module, $x \in X$, and $U$ a Stein open containing $x$. Write $\mathfrak{m}_{x}$ for the maximal ideal of the stalk $\mathscr{O}_{X,x}$.

Question:

If the local cohomology supported at $\mathfrak{m}_{x}$, namely $H_{\mathfrak{m}_{x}}^{i}(M_{x})$, is nonzero, can I conclude the cohomology with support in $x$, namely $H_{x}^{i}(U, M)$, is nonzero?

What about the reverse implication?

In the algebraic affine setting I believe the answer is yes, but I am naive to the vagaries of analytic objects.