Conjecture that the spheres $L_2$ and $L_3$ are non-polytopal

Let $L_2$ and $L_3$ be the combinatorial spheres obtained as the indicated vertex links in the constructed combinatorial manifolds.

Non-polytopality conjecture. The combinatorial spheres $L_2$ and $L_3$ are non-polytopal.

The paper proves that $L_1$ and $L_4$ are unflippable and hence non-polytopal, while $L_2$ and $L_3$ admit bistellar $8$-moves. Thus the latter claim is not implied by the unflippability argument and remains open in the source.