The spanning conjecture for complete cd-indices of Bruhat intervals

Let $n>0$. A Bruhat interval of rank $n+1$ is an interval $[u,v]$ in a Coxeter group with $\ell(v)-\ell(u)=n+1$. Its complete $cd$-index is the associated noncommutative polynomial in $c$ and $d$. The relevant space consists of $cd$-polynomials of degree at most $n$ whose nonzero homogeneous components all have degree congruent to $n$ modulo $2$.

Spanning conjecture. For all $n>0$, the complete $cd$-indices of all Bruhat intervals of rank $n+1$ span the whole space of $cd$-polynomials of degree bounded by $n$ whose nonzero homogeneous components have degree of the same parity as $n$.

The conjecture was verified computationally for $n\leq 17$. It would establish that complete $cd$-indices of Bruhat intervals provide all possible elements of the indicated parity-graded spaces and implies the absence of the universal relations described in the second candidate.