Uniqueness conjecture for TEP-structures over generically semisimple F-manifolds

Let $((M,0),\circ,e,E)$ be an irreducible germ of a generically semisimple $F$-manifold, meaning that the multiplication is semisimple at generic points, and suppose that $(T_0M,\circ)$ is a Gorenstein ring. A $(TEP)(w)$-structure is a structure of the type considered above the $F$-manifold, with weight $w$. Uniqueness conjecture. Above $((M,0),\circ,e,E)$, there exists, up to isomorphism, a unique $(TEP)(w)$-structure. Its only automorphisms are $\pm\operatorname{id}$. The conjecture concerns irreducible germs that are generically, but not necessarily everywhere, semisimple; it asserts both uniqueness up to isomorphism and the absence of automorphisms beyond the two scalar signs. The source indicates that this was intended for future work, and provides no resolution.