As far as I can find, this is left undefined
The rules do not actually define clearly what a “Strength-based skill” or “Dexterity-based skill” is in the face of things that change which ability score a skill uses. I can find nothing that says if Acrobatics, for example, is always a Dexterity-based skill even if it does not use Dexterity.
I similarly can find nothing that says that Acrobatics is a Dexterity-based skill only if it uses Dexterity, and that it would switch to being a Wisdom-based skill because it uses Wisdom.
Thus, we have an ambiguity, and I do not think there is any official rule resolving it.
In any event, I’m also somewhat dubious on the implied “requirement” that a skill be Strength-or-Dexterity-based in order to be affected by Armor Check Penalty; I tend to see that particular rule as a statement of how things happen to be (all ACP-affected skills are, by default, Strength-or-Dexterity-based) rather than a rule about how things must be (Strength-or-Dexterity-based as a requirement for being ACP-affected).
Thus, I’d be fairly leery of stating a hard-and-fast rule that skills are always “based” on their usual ability, or on their current ability, because it seems to me that if this is ambiguous in the rules, the designers may have had conflicting ideas on it when designing abilities. Some things probably just work better, or make more sense, if they treat Acrobatics as Wisdom-based when you have this trait. As a DM, I wouldn’t want to set any precedent.
From that perspective, I’d probably rule that it continues to be affected by Armor Check Penalty. And then I’d probably rule it as a Dexterity-based or Wisdom-based skill on a case-by-case basis, for things that specifically care about it being one or the other. But that’s getting quite far from RAW now.