Let $F_0\subset F_1\subset\cdots\subset F_n$ be a series of finite extension of fields. Suppose that $E$ is an intermediate field of $F_n/F_0$, do we necessarily have $$ \max_{1\le i\le n}[E\cap F_i:E\cap F_{i-1}]\le\max_{1\le i\le n}[F_i:F_{i-1}]?\tag{*} $$ Note that $[E\cap F_i:E\cap F_{i-1}]\le\max_{1\le i\le n}[F_i:F_{i-1}]$ for every $1\le i\le n$ does not need to hold, as shown by this example: Let $\alpha$ satisfy an irreducible cubic $f$ over $\mathbb{Q}$, such that $\mathbb{Q}(\alpha)$ is not Galois over $\mathbb{Q}$; let $\alpha'$ be another root of $f$. Suppose that $F_0=\mathbb{Q}$, $F_1=\mathbb{Q}(\alpha)$, $F_2=\mathbb{Q}(\alpha,\alpha')$ and $E=\mathbb{Q}(\alpha')$, then $[F_1:F_0]=3$, $[F_2:F_1]=2$, $[E\cap F_1:E\cap F_0]=2$, $[E\cap F_2:E\cap F_1]=3$. Also, in the case where $[F_i:F_{i-1}]=2$ for all $i$, $F_n$ is called an iterated quadratic extension of $F_0$, so $(*)$ would imply that a subextension of an iterated quadratic extension is again an iterated quadratic extension.
Can anything be said with regard to $(*)$? Thank you for any help.