The non-associative Moufang loop conjecture for cubic surfaces over number fields

Let $V \subset \mathbb{P}^3_L$ be a smooth cubic surface defined over a number field $L$. Let $\nu_1$ and $\nu_2$ be two different places of $L$. Suppose that there exists a non-associative Moufang loop of point classes on $V(L_{\nu_1})$, and let $\mathcal{O}$ be the ring of integers of $L_{\nu_2}$ with maximal ideal generated by a uniformizing element $\pi$. Let $V_{\nu_2}$ be the Zariski closure of $V$ in $\mathbb{P}^3_{L_{\nu_2}}$, and let $\tilde V_{\nu_2}$ be its special fibre. Assume that $\tilde V_{\nu_2}$ is a cone over a smooth cubic curve over $\mathcal{O} \mathbin{\bmod} \pi$. An admissible equivalence $\mathcal{A}$ is an equivalence relation on $V(L)$ compatible with the collinearity operation on point classes. The conjecture. Under these conditions, there exists an admissible equivalence $\mathcal{A}$ on $V(L)$ such that the Moufang loop $V(L)/\mathcal{A}$ is not associative. The first condition can occur only when the residue field of $L_{\nu_1}$ has characteristic $3$, while the conjecture predicts that the local non-associativity and the specified cone reduction at a second place produce a global non-associative quotient. The source provides no resolution of this conjecture.