Locally flat arcs in the four-dimensional product are unknotted

In those collars the image of $\alpha$ is vertical, so write $\alpha (t)=(p,h_0(t))$ near zero and $\alpha (t)=(p,h_1(t))$ near one. Each $h_i$ is an increasing homeomorphism onto its height interval: it is continuous and injective, and the endpoint heights fix its orientation. Choose an increasing homeomorphism $\theta :[0,1]\to [0,1]$ agreeing with the two inverse height functions near the endpoints and interpolate monotonically between them on the remaining middle interval. Then $\beta (s)=\alpha (\theta (s))$ has the same image as $\alpha$ and satisfies $\beta (s)=(p,s)$ on neighborhoods of the source endpoints. In the additional parametrized case take $\theta =\mathrm{id}$. This changes only the parameter used to describe the input image.

Apply [F1] to $\beta$, obtaining an ambient topological isotopy $A_t$ whose final arc $\gamma (s)=A_1(\beta (s))$ is smoothly embedded. Since $A_t$ is fixed near the boundary levels, $\gamma (s)=(p,s)$ near its endpoints. Apply the parametrized clause of [F2] to $\gamma$, obtaining an ambient smooth isotopy $B_t$ fixed near the boundary levels and satisfying $B_1(\gamma (s))=(p,s)$ for every $s$.

Concatenate $A_{2t}$ for $0\le t\le 1\mathrm{/}2$ with $B_{2t-1}{A}_1$ for $1\mathrm{/}2\le t\le 1$. They agree at the joining time. This is a continuous ambient family of homeomorphisms starting at the identity, fixed on a common neighborhood of the two boundary levels. Its terminal map sends the image of $\beta$, and hence the image of $\alpha$, to $\{p\}\times [0,1]$. In the additional parametrized case $\theta =\mathrm{id}$, so it sends $\alpha (s)$ to $(p,s)$ for every $s$. This proves both conclusions without an added choice hypothesis.