Near-extremal intrinsic functional conjecture for hyperbolic 3-manifolds

Let $\Psi$ be the collection of closed or cusped orientable hyperbolic $3$-manifolds, and define

$$D_i(M)=\inf_{\alpha\in\mathcal{H}^1}\frac{\pi\|\alpha\|_{Th}}{\sqrt{\operatorname{vol}(M)}\|\alpha\|_{L^2}}.$$

Here $\mathcal{H}^1$ denotes the space of harmonic $1$-forms, and $\|\cdot\|_{Th}$ and $\|\cdot\|_{L^2}$ are the Thurston and $L^2$ norms.

Near-extremal intrinsic functional conjecture. There exists an $\epsilon>0$ such that

$$\{M\mid D_i(M)>1-\epsilon\}=\{\text{fibered hyperbolic $3$-manifolds with some special geometry}\},$$

and there exists a sequence of closed manifolds $M_i$ such that $D_i(M_i)\rightarrow1$.

The conjecture seeks to characterize manifolds whose intrinsic functional is close to its upper bound and asks whether closed hyperbolic $3$-manifolds can approach that bound. The supplied text does not identify the special geometry or give evidence of a resolution.