Exceptional-class conjecture for symplectic embeddings into integral polydiscs

Let $c_b(a)$ be the infimum of the $\lambda>0$ for which the ellipsoid $E(1,a)$ symplectically embeds into the polydisc $P(\lambda,\lambda b)$, and let

$$E_n=\left(n,1;1^{\times(2n+1)}\right),\qquad n=b,\ldots,b+\left\lfloor\sqrt{2b}\right\rfloor,$$ $$F_b=\left(b(b+1),b+1;b+1,b^{\times(2b+3)}\right).$$

The associated exceptional classes give embedding obstructions. Exceptional-class conjecture. For every real $b\geqslant 2$, the function $c_b(a)$ is the maximum of the volume constraint

$$\sqrt{\frac{a}{2b}}$$

and the obstructions arising from the exceptional classes $E_n$ and $F_n$ in the displayed family. This would identify the listed exceptional classes as a complete set of obstructions for all real $b\geqslant2$. The surrounding text says that the claim is motivated by the theorem and related investigations, but supplies no proof or resolution.