Exponential decay conjecture for the crosscap-number defect

For a prime alternating knot or link $K$, let $\gamma(K)$ denote its crosscap number and let $\Gamma(K)$ denote its unoriented genus. Among all prime alternating knots, let $\varepsilon_K(c)$ be the proportion with $\gamma(K)>\Gamma(K)$ at crossing number $c$; similarly, let $\varepsilon_L(c)$ be this proportion among prime alternating links.

Exponential decay conjecture. Both proportions exponentially decay with respect to crossing number:

$$\varepsilon_K(c) \propto e^{-c\lambda_K} \qquad\text{and}\qquad \varepsilon_L(c) \propto e^{-c\lambda_L},$$

for positive constants $\lambda_K$ and $\lambda_L$. In particular, both proportions approach zero as the crossing number tends to infinity.

The conjecture is motivated by log-linear plots of the observed proportions and remains open.