A classical theorem of Goldstine is the following: Let $X$ be a Banach space and $J \colon X \to X''$ the natural inclusion. Then $J(B_X)$ is $\sigma(X'', X')$-dense in $B_{X''}$, where $B_Y$ is the unit ball in a space $Y$.

Now my question is if a related result holds when $X$ is only a quasi-Banach space, say if we impose some richness conditions for the dual spaces.