The sum-rule conjecture for two-boundary Temperley–Lieb ground states

Let $si$ be the ground-state vector, and let $S^{\mathrm{X}}_{N}(q,Q)$ denote the sum of all its components on the Kazhdan–Lusztig basis of boundary type $\mathrm{X}$, where $\mathrm{X}\in\{\mathrm{A},\mathrm{BI},\mathrm{BII},\mathrm{BIII}\}$. Let $A_n$ count symmetric binary matrices with no row sum greater than one, let $B_n$ count bisymmetric permutation matrices modulo rotation by $\pi/2$, and let $C_n$ be the signed-permutation sequence described in the source. Sum-rule conjecture. At $q=1$ and $Q=1$,

$$S^{\mathrm{A}}_{N}=A_N,\qquad S^{\mathrm{BI}}_{N}=B_{N+1}\quad\text{for }M=1,$$ $$S^{\mathrm{BI}}_{N}=C_{N+1}\quad\text{for }M=\infty,\qquad S^{\mathrm{BIII}}_{N}=C_{N+1}.$$

These identities conjecturally identify the total ground-state weights with the stated matrix enumerations; the paper gives recurrences and initial values for the sequences and notes the BI/BIII coincidence at $M=\infty$, but no resolution is supplied.