The conjecture on Diamond and Trapezoid 2×3×3 Abalone constellations

Let $C$ be a $2\times 3\times 3$ Abalone constellation, and call it a member of the “Diamond” or “Trapezoid” family according to the indicated subconfiguration. Let $o(C)$ denote its outcome. Diamond and Trapezoid constellation conjecture.

$$o(C)=\mathcal{L}.$$

The claim concerns two explicitly illustrated families of subconfigurations in $2\times 3\times 3$ Abalone; the supplied text gives no evidence that it has been proved or refuted.