Conjecture that the compatibility density of delta-matroids tends to zero

For each $n\geq 2$, let $\Gamma_n$ be the proportion defined in the paper for compatible pairs of labelled delta-matroids. The sequence $(\Gamma_n)_{n\geq 2}$ is decreasing and bounded below by zero, so its limit exists. Compatibility-density conjecture.

$$\Gamma_n \to 0 \quad \text{as } n \to \infty.$$

The conjecture is motivated by the observed values $\Gamma_2=1$, $\Gamma_3 \simeq 0.865$, $\Gamma_4 \simeq 0.649$, $\Gamma_5 \simeq 0.476$, and $\Gamma_6 \simeq 0.369$, together with the authors' inability to find larger classes of delta-matroids than those used for their lower bound. No resolution is given in the source.