The genus-2 extremal metric conjecture

Let $M$ be a surface of genus $2$, with Laplace eigenvalue $\lambda_1$ and area $\operatorname{Area}(M)$. The known upper bound in genus $2$ is

$$\lambda_1\operatorname{Area}(M)\leq 16\pi.$$

The genus-2 extremal metric conjecture. There exists a metric on a surface of genus $2$ that attains the upper bound

$$\lambda_1\operatorname{Area}(M)=16\pi.$$

The conjecture seeks sharpness of the genus-2 eigenvalue bound and is described as work in progress in the cited paper; the source gives no resolution.