Universal clique-spectral extremal conjecture for odd-order trees

Determine, for every fixed k≥2 and r≥2 and all sufficiently large n, whether the universal r-clique spectral extremal quantity for graphs on n vertices that omit at least one tree of order 2k+3 satisfies Λ_r(n,2k+3)=ρ_r(S_{n,k}^{+}) for 2≤r≤k+1 and Λ_r(n,2k+3)=ρ_r(K_{2k+2})=\binom{2k+1}{r−1} for k+2≤r≤2k+2; additionally, establish uniqueness of S_{n,k}^{+} in the first range and show that every extremal graph in the second range contains K_{2k+2} as a component.

Background

For odd tree order, the conjectured extremal transition is between the join-type graph S_{n,k}^{+}, which contains one additional edge in its independent part, and the complete component K_{2k+2}. The conjecture specifies both the exact spectral values and the extremal-graph structure in the two clique-order ranges.