Unimodality conjecture for Hamiltonian circle actions with isolated fixed points

Let $(M,\omega)$ be a $2n$-dimensional closed symplectic manifold equipped with a Hamiltonian $S^1$-action whose fixed points are all isolated. Write $b_j$ for the $j$-th Betti number of $M$. Unimodality conjecture. The even Betti numbers are unimodal, namely

$$b_{2i} \leq b_{2i+2}$$

for every $0 \leq i < \left\lfloor\frac{n}{2}\right\rfloor$. This asks whether Hamiltonian circle actions with isolated fixed points force the same monotonicity of even Betti numbers that holds for closed Kähler manifolds; the supplied source gives no resolution, so the conjecture remains open.