On the inertia set of a signed tree with loops

A signed graph is a pair (G,Σ)(G,Σ), where G=(V,E)G=(V,E) is a graph (in which parallel edges and loops are permitted) with V={1,…,n}V={1,…,n} and Σ⊆EΣ⊆E. The edges in Σ are called odd edges and the other edges of E   even. By S(G,Σ)S(G,Σ) we denote the set of all n×nn×n real symmetric matrices A=[ai,j]A=[ai,j] such that if ai,j<0ai,j<0, then among the edges connecting i and j  , there must be at least one even edge; if ai,j>0ai,j>0, then among the edges connecting i and j  , there must be at least one odd edge; and if ai,j=0ai,j=0, then either there must be at least one odd edge and at least one even edge connecting i and j, or there are no edges connecting i and j  . (Here we allow i=ji=j.) For a real symmetric matrix A, the partial inertia of A   is the pair (p,q)(p,q), where p and q are the number of positive and negative eigenvalues of A  , respectively. If (G,Σ)(G,Σ) is a signed graph, we define the inertia set of (G,Σ)(G,Σ) as the set of the partial inertias of all matrices A∈S(G,Σ)A∈S(G,Σ). By MR(G,Σ)MR(G,Σ) we denote max⁡{rank(A)|A∈S(G,Σ)}max⁡{rank(A)|A∈S(G,Σ)}. We say that a signed graph (G,Σ)(G,Σ) satisfies the Northeast Property if for each (p,q)(p,q) with p+q