The arbitrary modal-combination derivability conjecture

Let $\Delta$, $\nabla$, $\bullet$, and $\circ$ denote the contingency, accident, and their dual modal operators in the bimodal logic under consideration. For $m\in\mathbb{N}$, let $\heartsuit$ be any combination of $\Delta$, $\nabla$, $\bullet$, $\circ$, and negation. Arbitrary modal-combination conjecture. The logic should derive

$$\vdash\Delta\varphi\to\Delta^m\heartsuit\varphi$$

for all $m\in\mathbb{N}$ such that $m\geq 1$. This is a conjectural generalization of the preceding derivability claim to arbitrary combinations of the listed operators; the supplied text gives no resolution status beyond presenting it as a conjecture.