Odd-homology thickness conjecture for algebraic links with a positive tail

Let $p_i$ and $q_j$ be positive alternating rational tangles, with $m\ge2$, $n\ge2$, and let $+k$ denote a sequence of $k$ pluses, where $k\ge1$. Consider the algebraic link $(p_1,\ldots,p_m+k)-(q_1,\ldots,q_n)$. Let $\overline p_i$ and $\overline q_j$ denote the reduced tangles and set $q=\min(\overline q_1,\ldots,\overline q_n)$.

Positive-tail thickness conjecture. The link is $\overline{Kh}'$-thick if $q>1$ and either

$$\min(\overline p_1,\overline p_2,\ldots,\overline p_m)>1,\qquad k<q,$$

or

$$\min(\overline p_1,\overline p_2,\ldots,\overline p_m)=1,\qquad k+1<q.$$

This is the next criterion in the paper's computational analysis of algebraic links. The supplied text gives no resolution status.