The positivity conjecture for hyperbolic formal root coefficients

Let $v\le w$, let

$$\Phi^+\cap w\Phi^- = \\{\beta_1,\ldots,\beta_l\\},$$

and set $z_i=y_{-\beta_i}$. Let $I_{vw_\circ}$ be a reduced word as above, and let $\ell(v)$ denote the length of $v$. The positivity conjecture. (1) If $t=0$, then $\zeta_{I_{vw_\circ}}(ww_\circ)$ is a sum with positive coefficients of terms

$$u^{(m-\ell(v))/2}z_{j_1}\cdots z_{j_m},$$

where $\ell(v)\le m\le l$, $m-\ell(v)$ is even, and $1\le j_1<\cdots<j_m\le l$. (2) If $t=1$, then $\zeta_{I_{vw_\circ}}(ww_\circ)$ is a sum with positive coefficients of terms

$$(-1)^{k-\ell(v)}u^{(m-k)/2}z_{j_1}\cdots z_{j_m},$$

where $\ell(v)\le k\le m\le l$, $m-k$ is even, and $1\le j_1<\cdots<j_m\le l$. The two assertions are proposed from analogies with the ordinary cohomology and $K$-theory formulas and from experimental evidence. They concern positivity at the Lorentz specialization $t=0$ and the $K$-theoretic specialization $t=1$; no resolution is supplied in the given text.