Anisotropy conjecture for the middle socle of local face modules

Let $\sigma \\: \Gamma \to 2^V$ be a vertex-induced triangulation of a $(d-1)$-dimensional simplex, and let $k$ be a field. Let $B$ be the symmetric bilinear form on the local face module, and let $\operatorname{Soc} L^{d/2}(\Gamma)$ denote the middle-degree socle. Anisotropy conjecture. If $d$ is even, then the restriction of $B$ to $\operatorname{Soc} L^{d/2}(\Gamma)$ is anisotropic. Although $B$ need not be anisotropic on all of $L^{d/2}(\Gamma)$, even for regular triangulations over $\mathbb{Q}$, the asserted restriction remains an open question.