Let $ F $, $ F_i $ for $ i = 1, 2 $ and $ E $ be function fields over a finite field with characteristic $ p $ such that $ F \subseteq F_i \subseteq E $ and $ E = F_1 \cdot F_2 $ is the composite field of $ F_1 $ and $ F_2 $. Let $ Q $ be a place in $ E $, $ P_i = Q \cap F_i $ and $ P = Q \cap F $. Assume that $ P_i / P $ is wildly ramified for $ i = 1, 2 $. Is the following claim true? The ramification index $ e( Q | P_1 ) $ a divisor of $ e( P_2 | P ) $.

Why the question? First, by Abhyankar's Lemma, the claim is true if the extension $ P_i / P $ is tamely ramified for some $ i = 1, 2 $. Second, by using completions, the estimate $ e( Q | P_1 ) \le e( P_2 | P ) $ can at least be proven. Third, this estimate implies the claim for linearly disjoint Galois extensions $ F_1 / F $ and $ F_2 / F $ such that their degrees are powers of $ p $.

Anyway, I have the feeling that the claim is false. I looked in the proof of Abhyankar's Lemma in the book of Stichtenoth "Algebraic Function Fields and Codes" and I don't see how it could be modified to prove the claim. Also, a google search only yield that the claim is true for very specific Galois extensions.

Thanks