Let $D_1$ and $D_2$ be two smoothly embedded disks in $\mathbb{R}^3$ such that $\partial D_1 = \partial D_2$ and both disks intersect transversely on their interiors. Then there exists a $C\subset D_1\cap D_2$ such that $C$ bounds an inner most disk in both $D_1$ and $D_2$.
I believe this is true but I cant figure out a way to prove it for any pair of disks. Most papers which use "a standard inner most disk argument" dont provide any insight into why this should be true. A reference or suggestion would be appreciated.
Let $D_b$ and $D_r$ (for black and red disks) be the two disks obtained by rotating the picture below about the y-axis. As you spin the figure about the y-axis, the inner most circle in $D_b$ is the boundary for an annulus in $D_r$ and similarly for the inner most circle for $D_r$.