Conjecture on the limiting distribution of normalized Alexander-polynomial logarithms

Let $K$ be the knot produced by the interface curve in a three-dimensional three-colour percolation cube of size $N$, and let $\Delta_K$ denote its Alexander polynomial. Define

$$Y_N=\log\left|\Delta_K(-1)\right|,\qquad Z_N=\log\left|\Delta_K(i)\right|.$$

For $t\in\{-1,i\}$, write $\widehat{Y}_N$ and $\widehat{Z}_N$ for the corresponding normalized random variables, and define

$$h(x,\alpha)=1-\min\left(1,\left(\frac{(1-\alpha)x}{2-\alpha}\right)^{1-\alpha}\right), \qquad x\in\mathbb{R}_+,\ \alpha\in(0,1).$$

Alexander-invariant limit conjecture. For $t=-1$ (respectively, $t=i$), there exist numbers $c_t$, $\gamma_t$, and $\alpha_t$ such that the sequence of random variables

$$\frac{\log\left|\Delta_K(t)\right|}{c_tN^{\gamma_t}}$$

converges in law to a random variable with repartition function $h(x,\alpha_t)$. Approximate values are $\gamma_{-1}\simeq3.33$, $\alpha_{-1}\simeq0.44$, and $\alpha_i\simeq0.46$. The function $h(x,\alpha)$ is the repartition function associated with a density proportional to $x^{-\alpha}$ on a bounded interval, normalized to have mean $1$; the numerical fits are suggestive, but finite-size effects are too large to determine whether the exponent $\alpha$ is common to both evaluations or differs between them.