The maximal quadratic matrix equation conjecture in dimension three

Let $A_i$, $B_j$, $C_k$, $D$, and $F$ be generic $3\times3$ matrices, and let $X\in\mathcal{M}_3(\overline{K})$. Define

$$\psi(X)=A_1XB_1X+A_2XB_2X+XB_3XC_1+XB_4XC_2+XB_5X.$$

The maximal quadratic equation conjecture. The equation $\psi(X)=0_3$ has no nonzero solutions; equivalently, $X=0_3$ has multiplicity $512$. Moreover, the equation

$$\psi(X)+DX+F=0_3$$

has solution count and Galois group $(512,S_{512})$. Numerical experiments motivate both assertions, but the supplied text does not establish them.