Pretransitivity conjecture for modal interpretations

A normal modal logic $\Lambda$ is pretransitive if there is some $n\geq 1$ such that

$$\Lambda\vdash\Diamond^{n+1}p\to\bigvee_{k\leq n}\Diamond^kp.$$

Pretransitivity conjecture. If a normal modal logic $\Lambda$ is interpretable by a modal-to-modal translation with parameters in a pretransitive normal logic, then $\Lambda$ is pretransitive.

This conjecture proposes a necessary condition for interpretability between normal modal logics and would, for example, constrain interpretations from transitive or pretransitive logics. The supplied text does not indicate whether it has been resolved.