Local spectral rigidity of Liouville metrics on the torus

Let $g_0$ be a Liouville metric on $\mathbb{T}^d$, let $r$ be the regularity exponent appearing in the $C^r$ norm, and let $g$ be another Riemannian metric. Say that $g$ is Laplace isospectral to $g_0$ when their Laplace spectra agree. Local spectral-rigidity conjecture. There exists $\delta>0$ such that if

$$\|g_0-g\|_{C^r}<\delta$$

and $g$ is Laplace isospectral to $g_0$, then $g$ is isometric to $g_0$. This is proposed as a future nondeformational extension of the paper's rigidity theorem and remains open.