Suppose I am given a smoothly slice knot $K\subset\Bbb S^3$. But I am only given a locally flat disc $D\subset \Bbb D^4$ with boundary $K$.

Question:Is there a smooth disc $D'\subset\Bbb D^4$ with boundary $K$ that is $\epsilon$-close to $D$ and approaches $D$ towards the boundary?

More precisely, I want $D'$ to be in $D_\epsilon\cap X$, where $D_\epsilon$ is an $\epsilon$-thickening of $D$ and

$$X:=\bigcup_{x\in D} B_x(d(x,K)),$$

$B_x(r)$ being a ball around $x$ of radius $r$, and $d(x,K)$ being the distance of $x$ from $K$.