Let $S_1$ and $S_2$ be closed orientable surfaces with some metric defined on each such that the diameters and areas of $S_1$ and $S_2$ are bounded above by a constant $D$ and the injectivity radii of $S_1$ and $S_2$ are bounded below by a constant $\epsilon$.

Suppose $f:S_1 \to S_2$ is an $L$-Lipschitz map which is homotopic to an embedding. Can the lengths of the tracks of a homotopy from the immersion $f$ to an embedding be bounded above by a constant that depends only on $D$, $\epsilon$, and $L$?

Edit: The original question assumed that the original map was an immersion, but this implies (as pointed out in the comments) that the map is already an embedding.