The conjecture on the distribution of writhe in random grid diagrams

Let $w$ denote the writhe of a knot represented by an $n\times n$ grid diagram. The distribution is considered for knots encoded by two $n$-permutations recording the order of the columns and rows. Writhe distribution conjecture. The writhe follows a distribution with trivial expected value and skewness, and with variance and kurtosis given by

$$\operatorname{Var}(w)=\frac{1}{18}n^2 \qquad\text{and}\qquad \kappa(w)=3.5.$$

The conjecture is motivated by numerical experiments on randomly generated grid diagrams, whose sample means are close to zero, variances are proportional to $n^2$, and kurtoses are approximately $3.5$.