Semi-splitting conjecture for symplectic fibre bundles over spheres

Let $(M,\omega)$ and $(N,\eta)$ be compact symplectic manifolds. Let $(E,\pi,i)$ be a smooth fibre bundle over $S^d$ with fibre $M \times N$, such that

$$i^*:H^*(E) \longrightarrow H^*(M \times N)$$

is surjective. Assume that for all sufficiently large $\lambda$ there is a fibrewise symplectic structure $\Omega^{(\lambda)}$ on $E$ satisfying

$$i^*(\Omega^{(\lambda)})=\lambda(\omega\times 1)+1\times\eta.$$

Semi-splitting conjecture. The deformation of $H^*(M \times N)$ determined by $H^*(E)$ is semi-split with respect to $H^*(M)$. This conjecture proposes a general cohomological splitting principle for symplectic fibre bundles, extending the preceding semi-splitting result. Its resolution is not supplied in the source context.