Conjectured proportionality factors for the two-boundary Temperley–Lieb eigenvector

Let $L$ be the system size, let $z_1,\ldots,z_L$ be spectral parameters, let $\zeta_0$ and $\zeta_L$ be boundary parameters, and let $k$ and $s$ denote the functions and parameter used in the recursion relations for the eigenvector components. The notation $\hat z_i$ means that $z_i$ is omitted. The proportionality factors are denoted by $p$, $r_0$, and $r_L$. Proportionality-factor conjecture. For general $L$,

$$p(z_i;z_1,\ldots,\hat z_i,\hat z_{i+1},\ldots,z_L)=k(z_i,\zeta_0)^2k(z_i,\zeta_L)^2\prod_{j\neq i,i+1}k(z_i,z_j)^4,$$ $$r_0(z_2,\ldots,z_L;\zeta_0)=-k(\zeta_0,\zeta_L)\prod_{i=2}^{L}k(\zeta_0,z_i)^2,$$ $$r_L(z_1,\ldots,z_{L-1};\zeta_L)=-s^2k(1/s\zeta_L,s\zeta_0)\prod_{i=1}^{L-1}k(1/s\zeta_L,sz_i)^2.$$

These formulas extend the proportionality factors found for small system sizes and are intended to complete the recursive description of the eigenvector.