Jp-speciality conjecture for minimal positive braids

A positive minimal $k$-braid is a positive braid on $k$ strands whose braid presentation is minimal in the sense used in the source; let $L$ be its link. A link is Jp-special if its Jones polynomial is non-alternating or has gaps.

Jp-speciality conjecture. All minimal positive braids are Jp-special.

The claim is part of the paper's computational classification of Jp-special links. The supplied text gives no resolution status.