Weak Loewner spectral dominance of the connection matrix over the Dirac matrix

Let $G$ be a finite abstract simplicial complex with $n$ sets. Let $L$ be the connection matrix and $D$ the Dirac matrix. For a self-adjoint matrix $A$ with eigenvalues $\lambda_1(A)\geq\cdots\geq\lambda_n(A)$, define the spectral sums

$$S_k(A)=\sum_{j=1}^k\lambda_j(A).$$

Weak Loewner spectral-dominance conjecture. The matrices $L$ and $D$ satisfy

$$S_k(L)\geq S_k(D)$$

for every $1\leq k\leq n$. The weak Loewner order compares self-adjoint matrices through their ordered spectral sums and is weaker than the usual Loewner order. The paper reports experimental evidence for this inequality but states that it has not yet been proved.