Type-one tree complement conjecture for multiplicity bipartition

Let $T=S^k_{m,n}$ be the tree formed by taking a path on $k$ vertices and adding $m$ leaves to one endpoint and $n$ leaves to the other, where $kin\{2,3,4\}$ and $m,n>1$. Write $q(G)$ for the minimum number of distinct eigenvalues of a graph $G$, and $MB(G)$ for its multiplicity bipartition. Type-one tree complement conjecture. Then $q(\overline{T})=2$ and $MB(\overline{T})=3$. The conjecture concerns a family of complements of type-one trees and asserts both the two-eigenvalue property and the specified multiplicity bipartition; the surrounding discussion presents these as open problems.