The combinatorial non-crossing-sums inequality

Let $k_1,k_2,k_3$ be odd natural numbers. For each $i=1,2,3$, let $y_{i,1},\ldots,y_{i,k_i}$ be distinct real numbers, and set $y_{i,0}=y_{i,k_i+1}=-\infty$. For two finite pairs of real numbers, call their sums non-crossing when the two sums of one choice and the two sums of the other choice do not alternate in order. Assume:

1. For each $1\le i\le3$ and $0\le p<q\le k_i$ of the same parity, the pairs $\{y_{i,p},y_{i,p+1}\}$ and $\{-y_{i,q},-y_{i,q+1}\}$ have non-crossing sums.
2. For $0\le p_i\le k_i$ of the same parity, the three pairs $\{y_{1,p_1},y_{1,p_1+1}\}$, $\{y_{2,p_2},y_{2,p_2+1}\}$, and $\{y_{3,p_3},y_{3,p_3+1}\}$ have non-crossing sums.

Combinatorial formulation conjecture. Then

$$\sum_{i=1}^3\sum_{j=1}^{k_i}(-1)^{j-1}y_{i,j}<0.$$

This is an almost purely combinatorial formulation of the symmetric area inequality. It is stated as an open conjecture in the paper.