Monotonicity conjecture for the three-variable violation sets

Let $\mathcal{I}_n^3$ denote the set of triples that fail the relevant inequality for exponent $n$. For every $n\in\mathbf{N}$, the sets satisfy

$$\mathcal{I}_n^3\subseteq\mathcal{I}_{n+1}^3.$$

Monotonicity conjecture. For any $n\in\mathbf{N}$, we have

$$\mathcal{I}_n^3\subseteq\mathcal{I}_{n+1}^3.$$

Equivalently, if a triple $(x,y,z)$ fails to satisfy the inequality for some $n$, then it also fails for every $n'>n$. The conjecture is motivated by the proofs of the cited results and supporting numerical evidence; no resolution is given in the source.