Mori–Matano Wulff-shape conjecture for spreading fronts

Let $K(\theta)$ be the directional spreading speed and define the Wulff shape by

$$\mathscr{W}=\bigcap_{0\le\theta<2\pi}\left\{(x,y)^\top\in\mathbb{R}^2\mid x\cos\theta+y\sin\theta\le K(\theta)\right\}.$$

Let $\hat{\mathscr{F}}$ be the convex hull of the Frank diagram, let $P_\theta=(\cos\theta,\sin\theta)^\top/K(\theta)$ be the corresponding point on the Frank plot, and set

$$\mathcal{S}=\{\theta\in[0,2\pi)\mid P_\theta\in\partial\hat{\mathscr{F}}\}.$$

Assume that the normalized speed $c_*$ is positive and that the planar front in the direction of $\bm{n}^\theta$ is spectrally stable for every $\theta\in\mathcal{S}$. Mori–Matano conjecture. The asymptotic shape of the spreading front of the bidomain Allen--Cahn equation is described by the Wulff shape $\mathscr{W}$. Mori and Matano proposed this conjecture as a connection between spectral stability of the relevant planar fronts and the large-time geometric shape of spreading fronts; the source does not report a proof or disproof.