Thread hanging on socket write: why ? and how to prevent it?

I'm running a server for my game.
I'm using a quite simple system where basicly, there's a dedicated thread for reading incoming data for each of the connections.
Writing is totally independant from reading and is done in a pool of threads constantly polling a que of write tasks.

Everything is working fine, even in the evening of week-ends where there are easily more than 500 connected players in the same time.
This means more than 500 threads, what doesn't seem to cause any major problem most of the time. No excess CPU or network bandwidth load.

Except in one major case where a single player having a bogus connection might block the whole system.
It indeed happens regularely that a thread stays blocked on socketOutputStream.write for a long time, maybe several hours, if I don't try to interrupt it or forcefully close the corresponding socket.

Hance, from there, two questions:

Question 1: In which case a write operation on a socket might block ?

If it's obvious for read operations that it blocks so long as there's no new data to read, I don't understand very well why a write can block, unless I'm trying to send more data than my whole network connection can handle in total.
In this later case, it's obvious that I have to wait and block for a while... but it's far from being the case. I have 200 Mbps bandwidth limit and I'm hardly reaching 1 or 2 Mbps at rush hours.
Exchanged data is very minimal: no sound, no images, no video are going through when playing the game; only simple text commands.

Question 2: In the Socket class, there's a method setSOTimeout (see doc at https://docs.oracle.com/en/java/javase/11/docs/api/java.base/java/net/Socket.html), which sets a maximum timeout for read operations.
If the socket doesn't read anything within the timeout, a specific exception is launched, giving me the opportunity to check if the connection is still active and responding. I'm already using this mechanism to kick out non-responding players based on this.
However, it only applies on read operations.

Is there something similar for write operations ? I can't find anything.

Some additional precisions:

• The server is running on a linux machine, with Java AdoptOpenJDK 11.0.6
• Players can connect via SSL or not (a ServerSocket and a SSLServerSocket are listening on different ports). Most of them (60%) do use SSL. I'm using the default SSL configuration with a letsencrypt certificate. Using SSL doesn't appear to be slower than non-SSL connections.
• It seems that I hadn't this problem with Java 8. I have it since I upgraded from 8 to 11.

Thank you for your answers.

Writing to a socket may block if you keep sending data but the other side refuses to read it.

The operating system maintains a buffer for outgoing data, in case data needs gets lost and has to be sent again. Data is cleared from the buffer when the other party confirms receiving it ("acknowledge"). If the other party stops acknowledging data but you keep sending, eventually the buffer fills up, and further attempts to write will block so that data isn't lost.

There's no timeout setting for writing to sockets that I'm aware of. Maybe you could add an "are you still there" message to your application protocol, and close the connection if you don't get a reply. Another option is moving from blocking socket IO to non-blocking, but that's a major undertaking.