Monotonicity conjecture for new KMP eigenvalues

Let $\Gamma$ be a hypergraph with non-negative weights. For every $k\in\mathbb{Z}_{\geq 1}$, let $\omega_k=\omega_k(\Gamma)$ denote the smallest new eigenvalue of the Laplacian $\mathcal{L}(\Gamma,\mathcal{KMP}_k)$, meaning the smallest eigenvalue arising in the new invariant subspace at level $k$. Monotonicity conjecture for new KMP eigenvalues.

$$\omega_1\leq\omega_3\leq\omega_5\leq\cdots\qquad\text{and}\qquad\omega_2\leq\omega_4\leq\omega_6\leq\cdots.$$

The inequalities are supported by computer simulations and generalize the proved result for hypergraphs supported on subsets of size $n-1$. The source gives no resolution in general.