Subset Inequality Problem

Question

Here’s a problem I have been looking at recently. I’m looking for feedback, references, and/or solutions. Thanks. FGS 3.13.23.

Given

1. A set $X$ of $6$ natural numbers $X = \{X_1, X_2, X_3, X_4, X_5, X_6 \}$ such that $X_1 > 0$ and $X_{i+1} > X_{i}$.
2. Let $L_2$ and $R_2$ be any subsets of $X$ having $2$ elements such that $L2 ∩ R2 = \{ \}$ (the empty set).
3. The true or false value of the equation $∑ L_2 ≤ ∑ R_2$ is known for each such pair of subsets $L_2$ and $R_2$ as defined in $2$.

Prove or Disprove

Let $L_3$ and $R_3$ be any $3$ element subsets of $X$ such that $L_3 ∩ R_3 = \{ \}$. Then the true or false value of each such $∑ L_3 ≤ ∑ R_3$ can be determined without doing any additions on the elements of $X$.

Additionally, if it can be shown that the subset inequality problem for $6$ element sets cannot be solved with zero additions, what is the minimum number of additions, call it $SIP_{min}(6)$, needed to solve the problem?

Generalization

The above can be generalized to sets having $n > 6$ elements. Then determining $SIP_{min}(6)$ would be the induction basis step and $SIP_{min}(n)$ would be the induction step of a nice theorem.

Note that the Subset Inequality Problem is only interesting if $n$ is an even number. If $n$ is odd then $SIP_{min}(n)=0$.