Unobstructed Lagrangian tori

Question

Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in H^1(L)$ satisfy $\mu_0(1) + \mu_1(b) + \mu_2(b,b) + \ldots $ is a multiple of the identity?) If the answer is "not necessarily", what is an example? Note that Fukaya-Oh-Ohta-Ono show that Lagrangian torus orbits in toric manifolds are weakly unobstructed.