The condensed three-braid Jones polynomial coefficient conjecture

Let $\beta$ be a condensed positive three-braid word, let $w=w(\beta)$, and write $r$ for the number of syllables in $\beta$. Define $V_{\widehat{\beta}}^*(t)$ and $V_{\widehat{\beta}}^{**}(t)$ by the displayed decomposition below. Here $\epsilon_n=(-1)^n$, $[P]_j$ denotes the coefficient of $t^j$ in a polynomial $P$, and an AC polynomial is one whose nonzero coefficients alternate in sign. A condensed three-braid has either no trivial syllables or one trivial syllable whose two adjacent syllables each have length at least three.

Condensed three-braid conjecture. One has

$$V_{\widehat{\beta}}^*(t)=\epsilon_w(1+t^2)+t^2V_{\widehat{\beta}}^{**}(t),$$

and the following properties hold: the sign of $[V_{\widehat{\beta}}^{**}]_j$ is $\epsilon_{r+1+j+w}$, so $V_{\widehat{\beta}}^{**}$ is an AC polynomial; if $\beta$ has no trivial syllables, then

$$\deg V_{\widehat{\beta}}^{**}(t)=w-r-1,\qquad [V_{\widehat{\beta}}^{**}]_{\max}=1;$$

if $\beta$ has one trivial syllable, then

$$\deg V_{\widehat{\beta}}^{**}(t)=w-r-2,\qquad [V_{\widehat{\beta}}^{**}]_{\max}=-1;$$

and

$$[V_{\widehat{\beta}}^{**}]_j\neq 0\quad\text{exactly when}\quad j\in[w-2-\deg V_{\widehat{\beta}}^{**}(t),\,\deg V_{\widehat{\beta}}^{**}(t)].$$

The claim gives a precise alternating-sign and support description for the residual part of the Jones polynomial of condensed positive three-braid closures. The supplied text does not establish whether this statement is proved or remains open.