Can I make a PL-embedding of a 2-disc into a smooth embedding, and also the other way around?

Question

I have a piece-wise linear (PL) embedding of a disc $D^2$ into $\Bbb R^4$. Now, I want to perform some operation on this disc, but this operation only works with smooth embeddings. So my hope is that I can transition to a smooth embedding, perform the operation, and transition back to a PL-disc.

My questions are therefore:

1. if $\phi:D^2\to\Bbb R^4$ is the PL-embedding, is there another embedding of $D^2$ in an $\epsilon$-neighborhood of $\phi(D^2)$, that agrees with $\phi$ on $\partial D^2$ and is smooth on the interior of $D^2$ (ideally also with curvature bounded linearly in the distance from $\partial D^2$)?
2. if $\psi:D^2\to\Bbb R^4$ is an embedding that is smooth on the interior of $D^2$ and PL on $\partial D^2$, is there a PL-embedding of $D^2$ in an $\epsilon$-neighborhood of $\psi(D^2)$ that agrees with $\psi$ on $\partial D^2$?