On the intersections of maximal subgroups of finite simple groups

Question

I am looking for a proof or a counterexample for the following proposition:

Let $G$ be a finite simple group and $M_{1}$ and $M_{2}$ be two maximal subgroups of $G$ with nontrivial intersection.Then every conjugate of $M_{1}$ and $M_{2}$ have nontrivial intersection, i.e. if $M_{1}\cap M_{2}\neq1$, then $M_{1}^{x}\cap M_{2}^{y}\neq1$, for each $x,y\in G$.

Answer 1

If $M_1=M_2=H$, this holds if and only if the primitive action of $G$ on $[G:H]$ is not of base-two. So every base-two action of $G$ is a counterexample. There are a lot of counterexamples (even if $H$ is soluble), such as

1. $G=A_p$ and $H=\operatorname{AGL}_1(p)\cap A_p$ with $p\ne 5,7,11,17,23$ a prime number.
2. $G=\operatorname{M}_{23}$ with $H=23{:}11$.
3. $G=A_9$ and $H=\operatorname{ASL}_2(3)$.
4. $G=\operatorname{PSL}_2(q)$ and $H=D_{q-1}$, where $q\ne 5,7,9,11$.