KAC-T1, KAC-T2, and KAC-T3 marking-chain questions

For the continuous-time chain on $\{0,1\}^k$ with generator $Q$ defined by

where boundary terms are omitted and $\oplus$ denotes addition modulo $2$, determine the corresponding properties of the true spectrum and eigenmodes of $-Q$ on the path $P_k$. In particular: (1) does the true spectrum contain paired levels whose splitting is of order $2^{-k/2}$, i.e. asymptotically $\Theta(2^{-k/2})$; and (2) do the true eigenmodes satisfy the sign rule exhibited by the variational modes, namely that the relevant coefficient changes sign at $\theta^*=\sqrt{2}-1$ for even levels and has no sign change for odd levels? The source labels the associated variational results KAC-T1, KAC-T2, and KAC-T3; the supplied material establishes the variational bounds and their strictness for $k\ge 4$ but leaves these true-spectrum and true-mode questions unresolved.