The tropical Lagrangian torus conjecture for Fano hypersurfaces

Let $X^n_a$ be the degree-$a$ Fano hypersurface in projective space with $a=n-1$, and let $L_n\subset X^n_a$ be the monotone Lagrangian torus obtained as the torus fibre over the centroid of the unique compact cell of the corresponding tropical manifold. Write $w(L_n)$ for its disk potential value, and let $C\!\ell_{n-2}$ denote the Clifford algebra on $n-2$ generators.

Tropical Lagrangian torus conjecture. The torus satisfies

$$w(L_n)=a^a-a!,$$

so that $L_n$ lies in the unique small component of the monotone Fukaya category, and

$$HF^*(L_n,L_n)\cong C\!\ell_{n-2}.$$

This proposes an explicit monotone Lagrangian representative of the small Fukaya-category component in the Fano index-one case. The statement is presented as a speculation about a possible construction, and no resolution is supplied in the source.