Fox's Trapezoidal Conjecture asserts that the coefficients of the Alexander polynomial of an alternating knot alternate and the sequence of their absolute values forms a trapezoidal shape.
The same is true for the Jones polynomial for some alternating knots that I checked. However, since I didn't find an analogous conjecture for the Jones polynomial, I expect that it is not true for some known knot. Which one?
Was there proposed perhaps some more narrow version above conjecture for the Jones polynomial of alternating knots? (Either with stricter assumptions or with relaxed conclusions.)
The alternating knot $8_{10}$ has Jones polynomial $$V_{8_{10}}(t) = -t^{-6}+ 2t^{-5}-4t^{-4}+ 5t^{-3}-4t^{-2}+ 5t^{-1}-3+ 2t-t^2,$$ which does not have trapezoidal coefficients.
I am not aware of an alternate version of the trapezoidal conjecture for the Jones polynomial.