Fix a prime $p$ and let $L$ and $K$ be two finite extensions of $\mathbb{Q}_p$ with ramification degrees $e_L$ and $e_K$, respectively. Let $e'$ and $e''$ be the ramification degrees of $LK/L$ and $LK/K$, respectively. Is it true that $e'\leq e_K$, or are there some additional constraints we can add to $L$ and $K$ which make this true? It is trivially true if $L=\mathbb{Q}_p$ (then $e'=e_K$), or if $K$ is unramified over $\mathbb{Q}_p$ (then $e_K=e'=1$), but I'm not exactly sure what to do if $L$ is strictly larger than $\mathbb{Q}_p$ and $K$ is ramified.
Clearly we have $e_Ke''=e_Le'$ by multiplicativity of ramification indices in towers, so the ratios $e_K/e'$ and $e_L/e''$ are equal. Intuitively, it seems like $e'\leq e_K$ should be true, in that by adjoining elements of $K$ to $L$ we should not end up with "any more" ramification than we began with by adjoining elements of $K$ to $\mathbb{Q}_p$. In other words, it would be nice to say that the ramification occuring in $LK/L$ is no worse than the ramification occuring in $K/\mathbb{Q}_p$. I can't seem to make this precise though, and am not sure whether it's true. Help or counterexamples would be much appreciated. I imagine the case where $K/\mathbb{Q}_p$ is tamely ramified might be the next easiest case, since then $LK/L$ is also tamely ramified, but I haven't had much success with that so far either.
$$e(K/F) = \frac{v(\pi_F)}{v(\pi_K)}$$
It is also $$[K:F(\zeta_{q_K-1})]$$ where $q_K$ is the cardinality of $O_K/(\pi_K)$.
Clearly $$[K:F(\zeta_{q_K-1})]\ge [LK:LF(\zeta_{q_{LK}-1})]$$