Conjecture on the balanced genus of products of spheres with the circle

Let $d\geq 3$, and let $\mathcal{G}_M$ denote the balanced genus of a PL $d$-manifold $M$. Consider the product manifold $\mathbb{S}^{d-1}\times\mathbb{S}^1$. Balanced-genus conjecture.

$$\mathcal{G}_{\mathbb{S}^{d-1}\times\mathbb{S}^1}=4+3(d-3)2^{d-2}.$$

This extends the calculated values $\mathcal{G}_{\mathbb{S}^2\times\mathbb{S}^1}=4$ and $\mathcal{G}_{\mathbb{S}^3\times\mathbb{S}^1}=16$ to higher dimensions. The conjecture concerns the balanced genus of a family of non-sphere PL manifolds and remains open.