Beraha-number vanishing-amplitude conjecture for triangular-lattice strips

Let $L$ be the width of a triangular-lattice strip with free, cylindrical or “zig-zag” boundary conditions. Let $B_2,\ldots,B_{L+1}$ be the corresponding Beraha numbers, let $\alpha_i(q)$ denote transfer-matrix amplitudes, and let $D(q)$ be the matrix whose determinant vanishes when at least one such amplitude vanishes. Beraha-number vanishing-amplitude conjecture. For a triangular-lattice strip of width $L$ with free, cylindrical or “zig-zag” boundary conditions, at each Beraha number $q=B_2,\ldots,B_{L+1}$ there is at least one vanishing amplitude $\alpha_i(q)$. That is, $\det D(q)=0$ for $q=B_2,\ldots,B_{L+1}$. The claim is motivated by empirical observations for all studied widths and extends an analogous conjecture for the square lattice; it is stated for arbitrary $L$ and remains open.