Gram-matrix parity conjecture for the dual canonical basis

Let $TL_M$ be the Temperley–Lieb algebra, let $\{t_i\}$ be the dual canonical basis vectors obtained from the corresponding algebra elements $a_i$, and let ${\Greekmath 0111}$ define the product on the representation space. Denote by $G$ their Gram matrix:

$$G_{ij}=\langle t_i,{\Greekmath 0111} t_j\rangle.$$

For an algebra element $a$, define

\limfunc{tr}a=\limfunc{tr}ae_M=\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{the number of closed loops}

obtained by closing the planar diagram associated with $a$.

Gram-matrix parity conjecture. One has

G_{ij}=0\qquad\relax\protect\ifmmode\expandafter\text@\else\expandafter\text\fi{whenever}\qquad \limfunc{tr}(a_i a_j)=0\pmod 2,

where $a_i$ and $a_j$ are the algebra elements corresponding to $t_i$ and $t_j$, respectively.

The conjecture is based on numerical computations for $M=3,5,7$ and predicts a parity-controlled vanishing pattern for the Gram matrix in the dual canonical basis. The supplied passage does not state a proof or a resolution, so the conjecture remains open.