Complex conjugate generalized-eigenvectors of a non-diagonalizable matrix

Question

Let $v_i$ ($i = 1,\ldots m$) be the generalized eigenvectors of matrix $A$ associated with the Jordan block $J(\lambda)$, where $\lambda$ is a complex eigenvalue of $A$.

Can we show that the complex conjugates $\bar{v}_i$ ($i = 1,\ldots m$) are the generalized eigenvectors of $A$ associated with $J(\bar{\lambda})$?

Thanks!