Odd-homology thickness conjecture for reduced Montesinos links

Consider a non-alternating Montesinos link with Conway symbol $p_1,\ldots,p_m,-q$, where the $p_i$ and $q$ are positive rational tangles as in the source. Reduce each rational tangle by replacing a positive tangle of length greater than one by its last number, a negative tangle by minus its last number minus one, and leaving length-one tangles unchanged; denote the resulting tangles by $\overline p_i$ and $\overline q$.

Montesinos thickness conjecture. A Montesinos link of the form $p_1,\ldots,p_m,-q$ is $\overline{Kh}'$-thick if it has a minimal-crossing Conway symbol satisfying

$$\min(\overline p_1,\overline p_2,\ldots,\overline p_m)\ge\overline q.$$

For links with at most $11$ crossings, the text says the computational results agree with this criterion, but it also notes exceptional thicker links for larger crossing numbers, so the unrestricted claim is not established.