The predicted unit signature-rank distribution for odd-degree S_n-fields

Let $K$ be a number field with signature $(r_1,r_2)$ whose Galois closure has Galois group $S_n$, where $n$ is odd. Let $E_K$ be the unit group of $K$, and let $\operatorname{sgnrk}(E_K)$ denote its signature rank. Define $(q)_m=\prod_{i=1}^m(1-q^{-i})$. Unit signature-rank conjecture. As $K$ varies over these fields, ordered by absolute discriminant, for every integer $s$ with $1\leq s\leq r_1$,

$$\operatorname{Prob}(\operatorname{sgnrk}(E_K)=s)=2^{(r_1+r_2-s)(s-r_1)}\frac{(2)_{r_1+r_2-1}(4)_{(r_1-1)/2}(4)_{(r_1-1)/2+r_2}(2)_\infty}{(2)_{r_1+r_2-s}(2)_{s-1}(4)_\infty}$$ $$\times\sum_{\rho=\max(0,(r_1+1)/2-s)}^\infty\frac{1}{2^{\rho(r_1+r_2-1)+\rho(\rho+1)/2}(2)_{r_1+r_2-1+\rho}} \sum_{k=\max(0,r_1-s-\rho)}^{\min(r_1-s,(r_1-1)/2)}\frac{2^{k(r_1-s-k)}(2)_{r_1-1-k}(2)_{\rho+k+r_2}}{(2)_{r_1-s-k}(4)_{(r_1-1)/2-k}(2)_k(2)_{k+r_2}(2)_{\rho+k-r_1+s}}.$$

This prediction is obtained by combining the conditional signature-rank distribution with the predicted class-group rank and rank-difference distributions. The source gives no resolution.