Hill climbing a password would only be possible if intermediate KV cache entries were stored. To hillclimb "hunter2", you're going to try "a", "b", "c", etc, until you notice that "h" comes back faster. Then you try "ha", "hb" and so on.

But that's only going to work if the cache looks like: "h", "hu", "hun", ..., "hunter2"

If just "hunter2" is in the cache, you won't get any signal until you stumble on exactly that password. And that's before getting into the block size granularity of the caches discussed elsewhere in this thread.

That's not to say timing attacks aren't possible. I haven't looked at Claude Code's prompt generation, but there's no intrinsic reason why you couldn't do things like figure out what open source code and research papers your competitors are loading into context.

Sharing caches between orgs would be an incredible misstep.