Highest-sector eigenvalue conjecture for triangular-lattice toroidal strips

Let $T_{L,L}$ be the diagonal transfer-matrix sub-block for a triangular-lattice strip of width $L$ with fully toroidal boundary conditions, and let $b^{(L)}_1$ denote the stated amplitude. Triangular-lattice highest-sector eigenvalue conjecture. The block $T_{L,L}$ contains the eigenvalue

$$\lambda=(-1)^L 2$$

with amplitude $b^{(L)}_1$. The source presents this as an analogue of the square-lattice highest-sector conjecture and gives no proof.