Conjecture that the stellated tetrahedron $\mathcal{P}_{11/20}$ is not Rupert

Let $\mathcal{P}_a$ be the stellated tetrahedron with vertices

$$p_0=(1,1,1),\quad q_0=(-a,-a,-a),$$ $$p_1=(1,-1,-1),\quad q_1=(-a,a,a),$$ $$p_2=(-1,1,-1),\quad q_2=(a,-a,a),$$ $$p_3=(-1,-1,1),\quad q_3=(a,a,-a),$$

where edges join all $p_i,p_j$ and $p_i,q_j$ for $i\ne j$, and $a\in(0.5,0.57)$. A polyhedron is Rupert if it can pass through itself under the projection-and-translation containment condition. Stellated-tetrahedron conjecture. The stellated tetrahedron $\mathcal{P}_{11/20}$ is not Rupert. Numerical searches have not found a Rupert passage for the relevant family, and the stated sufficient condition for being locally Rupert does not apply; whether this particular stellated tetrahedron is non-Rupert remains open.