Jp-speciality conjecture for alternating knots and links

A link $L$ has Jones polynomial $J(L)=\sum c_iq^{2i}$. The polynomial is alternating if $c_ic_j>0$ implies $j=i\mod 2$ and $c_ic_j<0$ implies $j\ne i\mod 2$; a link is Jp-special if its Jones polynomial is non-alternating or has gaps.

Jp-speciality conjecture. All alternating knots or links, except those belonging to the family given by Conway symbol $n$—the torus links $(2,n)$, $n\ge2$—are not Jp-special.

This conjecture concerns the relationship between alternation of a link and the sign and support pattern of its Jones polynomial. The supplied text gives no resolution status.