The conjecture that four twice-punctured tori suffice in hyperbolic knot exteriors

Let $K$ be a hyperbolic knot in $S^3$. A twice-punctured torus is an embedded torus with two punctures, and an integral slope is a slope on its boundary having integral slope parameter. The tori are pairwise disjoint, non-isotopic, and nested when each lies in the complement of the others in the specified nested arrangement.

Four-torus bound conjecture. There are at most four pairwise disjoint, non-isotopic, nested, embedded twice-punctured tori with an integral slope in the complement of $K$.

The paper proves the weaker universal bound of six such tori for every hyperbolic knot. The proposed improvement to four is motivated by examples with four non-isotopic twice-punctured tori, but its resolution is not established in the supplied text.