A separable state in $\mathcal{H}_{a}\otimes\mathcal{H}_{b}$ is given by
$$\rho_{s}=\sum_{\alpha,\beta}p(\alpha,\beta)|\alpha\rangle\!\langle\alpha|\otimes|\beta\rangle\!\langle\beta|.$$
Now, my question is, can what can we say about the existence of $\{|\alpha \rangle\}$ and $\{|\beta \rangle\}$ such that all of them are elements from a complete basis (non-unique) in individual subsystem? Whether they exist? or for what kind state they exists (for example, in case zero discord state-they exist, as pointed out by one of the commentators1)
A reason I think such a set of basis exist for all separable state is because separable state space is the convex hull of tesnor products of symmetric rank-$1$ projectors which are affinely independent (Caratheodory's theorem), thus by definifion, there exists a set of linearly independent vectors for each subsystem (mostly, nonunique), let's say those are $\{\alpha\rangle\}$ and $\{|\beta\rangle\}$. Surely, we can add few more vectors to both suitably to span the whole space!! Is it true? Any help is appreciated.