A convert is considered to be the child of Abraham when called up to the Torah - ben Avraham Avinu.
Does this still apply if the convert was born as a non-Jewish paternal descendant of Abraham? For example, if an Edomite or Ishmaelite converts, do their retain a halachic paternal lineage through Edom or Ishmael back to Abraham or are they reborn as direct children of Abraham?
In other words, does the principle of calling a ger as ben Avraham Avinu apply to all gerim universally, or only to gerim that were not already among Abraham's seed?
I do recognize that the majority, if not the vast majority, of converts were not born paternal descendants of Abraham, and that of those that are, very few have proof of their descent. I am asking about how things work in theory, or how they would work if everyone had accurate genealogies.