ssh/scp config file with proxycommand still asks for password

Question

My setup is like this:

[local machine] --> [university network] --> [local machine in university network]
   [machine1]   -->       [machine2]     -->              [machine3]

I have setup passwordless access with ssh-keygen. When I'm on machine1, I can type ssh user2@machine2 and I am promptly connected. I can move on from there to type ssh user3@machine3 and I'm in.

I now want to be able to type ssh machine3 on machine1 and directly connect the the third machine. I wrote the following .ssh/config file:

Host machine3
    Hostname machine3
    ProxyCommand ssh user2@machine2 -W %h:%p
    User user3
    IdentityFile ~/.ssh/id_rsa.pub
    IdentitiesOnly yes  

However, now I am always prompted for the password for user3. If I type it in, I'm connected, so the tunneling works. How can I skip typing in the password for user3?

Machine1 is Windows10 Pro (I'm using the ssh command in the power-shell). Machine2 runs Ubuntu 20.04.2 LTS (Focal Fossa) Machine3 runs Linux Mint 20.3 Una

Answer 1

As hinted in the comments, I mistakenly assumed, that I would connect with a "stop" on machine2. However, that's not the case. The solution is to copy the ssh public key both to machine2 and machine3 from machine1.