Degree-three generic list recovery for MRA with projection

Let $p\geq 3$ be odd, let $V=\mathbb{R}^p$ with the cyclic-shift action of $G=\mathbb{Z}/p\mathbb{Z}$, and let the projection identify opposite entries as in MRA with projection. MRA-with-projection conjecture. For every odd $p\geq 3$, generic list recovery is possible at degree $3$. This is supported by Jacobian-criterion computations through $p=21$; degree $2$ cannot suffice because projection loses information.