2nd factor prompt not shown

Type: Bug

I'm connecting to a system that uses 2-factor authentication - password followed by a one-time password from an authenticator app. After typing in the password, I don't get a prompt for the 2nd factor. If I click on "details" in the "Setting up SSH Host" message in the bottom right corner, it opens a terminal where I can see the prompt and type in the one-time password.

However, this install terminal persists after connecting, commands can't be typed into it so it's useless, and killing it closes the connection (I submitted a separate issue about that: #11181 ), so it would be nice to just get a prompt for the second factor that doesn't end up showing this space-wasting install terminal, as for the password.

Extension version: 0.26.0
VS Code version: Code 1.103.2 (6f17636121051a53c88d3e605c491d22af2ba755, 2025-08-20T16:45:34.255Z)
OS version: Windows_NT x64 10.0.26100
Modes:
Remote OS version: Linux x64 5.14.0-570.30.1.el9_6.x86_64

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Reproduce an SSH connection using password followed by a one-time password, then inspect the "Setting up SSH Host" flow and its details terminal. Confirm where the second-factor prompt is handled and that the connection completes without leaving a useless terminal open; the issue provides no source file or test entry point.