I've heard that every $n$-PDA when $n > 2$ is as powerful as $2$-PDA. Unfortunately every proof I'm able to find uses references to Turing Machines, which I haven't learned about yet. I'm sure there must exist an alternative proof, supposedly one that converts $n$-PDA to $n-1$-PDA, and then proceeds by induction, but I'm unable to find it. Any references, or hints are greatly appreciated.
What I tried: to simulate two stacks using just one, therefore going from $n$ to $n-1$. But it would also mean that $2$-PDA are as powerful as $1$-PDA, so it's clearly a wrong way.