Null-reduction after purification implies original inequality is holographic

Determine whether, for any pair of (0,1) matrices (L,R) in the set P that encodes a candidate holographic entropy inequality, the condition that for all pairs of parties i and j the null reduction on i of the purification on j of (L,R) yields a holographic entropy inequality necessarily implies that (L,R) itself is a holographic entropy inequality.

Background

The paper proves that the null reduction of any superbalanced holographic entropy inequality (HEI) is again an HEI, and provides counterexamples to the stronger conjecture that if all null reductions of an inequality are HEIs then the original inequality must be an HEI. Motivated by these results, the authors formulate a weaker conjecture that considers null reductions after varying the purifier, thereby increasing the number of null reductions examined.

They present numerical evidence in favor of this weaker statement for N = 5, exhaustively testing over 100,000 superbalanced non-HIQ candidates without finding counterexamples. Nevertheless, they caution that counterexamples could be rare, so the conjecture remains unresolved.