The torus first Dirac eigenvalue optimisation conjecture for the trivial spin structure

Let $(a,b)$ satisfy $0\leq a\leq 1/2$ and $a^2+b^2\geq 1$, let $\mathbb{T}^2=\mathbb{R}^2/\Gamma_{a,b}$ be the corresponding flat torus with flat metric $g_{a,b}$, and let $[g_{a,b}]$ be its conformal class. Write $S_0$ for the trivial spin structure and $\Lambda_1(\mathbb{T}^2,[g_{a,b}],S_0)$ for the conformal supremum of the first Dirac eigenvalue normalised by area. The torus first eigenvalue conjecture. For the trivial spin structure $S_0$ on the torus $\mathbb{T}^2$ one has

$$\Lambda_1(\mathbb{T}^2,[g_{a,b}],S_0)= \begin{cases} \dfrac{2\pi}{\sqrt{b}} & \text{if } b\geq\pi,\\ 2\sqrt{\pi} & \text{if } b\leq\pi. \end{cases}$$

In other words, either the minimiser is flat, or there is no smooth minimiser and a minimising sequence degenerates to a bubble. The claim extends the theorem proved for $b>2\pi$ and is motivated by the existence of smooth minimisers in the relevant regime.