Converse to the octahedral relation for Sato tau functions

Let $\tau(n_1,\ldots,n_s)=\langle g\mid n_1,\ldots,n_s\rangle$ be the discrete function defined for integer tuples with $\operatorname{deg}(\mathbf n)=\sum_i n_i=0$. For $\mathbf n\in\mathbb Z^s$ with $\operatorname{deg}(\mathbf n)=-2$ and indices $\alpha<\beta<\gamma<\delta$, the octahedral relation is

$$\tau(\mathbf n+e^\alpha+e^\beta)\tau(\mathbf n+e^\gamma+e^\delta)-\tau(\mathbf n+e^\alpha+e^\gamma)\tau(\mathbf n+e^\beta+e^\delta)+\tau(\mathbf n+e^\alpha+e^\delta)\tau(\mathbf n+e^\beta+e^\gamma)=0,$$

where $e^\alpha$ is the standard unit vector in the $\alpha$-th position. Converse to the octahedral relation. The octahedral relation implies that the function arises from the stated Sato tau-function construction, and conversely the construction is characterized by this relation. The source gives no further hypotheses or proof of this converse, so its precise scope and status require verification.