The 27-Pfaffian conjecture for the reciprocal catalecticant variety

Let $\mathrm{Cat}(2,3)$ be the catalecticant variety, let $\mathbb{P}\\,\mathrm{Cat}(2,3)^{-1}$ denote its reciprocal variety, and let a cubic Pfaffian mean the Pfaffian of a $7 \times 7$ skew-symmetric matrix whose entries are linear forms. 27-Pfaffian conjecture. The reciprocal variety $\mathbb{P}\\,\mathrm{Cat}(2,3)^{-1}$ is defined by exactly $27$ cubic equations which are Pfaffians of possibly different $7 \times 7$ skew-symmetric matrices. Although the codimension is compatible with a description using Pfaffians of an $8 \times 8$ skew-symmetric matrix, the source notes that its degree and number of cubic generators would disagree with the numerical results; the stated $27$-equation description remains conjectural.