Quadratic-generation conjecture for canonical-form and log canonical compactifications

Let $\mathcal{X}_{\mathrm{cf}}(3,6)$ be the closure of the variety parametrized by the canonical forms of the real regions of $\mathcal{X}(3,6)$, and let the log canonical compactification of $\mathcal{Y}(3,6)$ be embedded in $\mathbb{P}^{149}$. Quadratic-generation conjecture. The vanishing ideals of $\mathcal{X}_{\mathrm{cf}}(3,6)$ and of the log canonical compactification of $\mathcal{Y}(3,6)\subset \mathbb{P}^{149}$ are generated by quadratic polynomials. The computed quadratic generators show that each full ideal is a minimal prime of its respective ideal of quadratics, but verifying quadratic generation is stated to be beyond current computational methods.