Cromwell's minimal-genus composite-diagram conjecture

Let $D$ be a diagram of a composite knot $K=K_1\mathbin{\#}K_2$, where $g(D)$ is the diagram genus and $\widetilde g(K)$ is the knot invariant used in the source. Assume

$$g(D)=\widetilde g(K).$$

Cromwell's conjecture. Then $D$ is composite.

This conjecture is offered as evidence for the apparent lack of exotic composite weak genus two knots. The supplied text does not state that it has been proved or disproved.