Can you get Warp terminal as the integrated terminal in VS Code?

I followed the directions on Warp's documentation, but this only replaces the external terminal (it opens as a separate window). https://www.warp.dev/blog/how-to-open-warp-vscode

I would like Warp to replace vscode's usage of xterm in the integrated terminal, but I can't seem to figure out how to do that. My attempt was to edit the profiles in my settings.json:

"terminal.integrated.profiles.osx": {
        "bash": {
            "path": "bash",
            "args": [
                "-l"
            ],
            "icon": "terminal-bash"
        },
        "zsh": {
            "path": "zsh",
            "args": [
                "-l"
            ]
        },
        "fish": {
            "path": "fish",
            "args": [
                "-l"
            ]
        },
        "tmux": {
            "path": "tmux",
            "icon": "terminal-tmux"
        },
        "pwsh": {
            "path": "pwsh",
            "icon": "terminal-powershell"
        },
        "warp": {
            "path": "/Applications/Warp.app/Contents/MacOS/stable",
            "icon": "/Applications/Warp.app/Contents/Resources/Warp.icns"
        }
    },
    "terminal.integrated.defaultProfile.osx": "warp",

but all this did was open up Warp in a separate window and the existing integrated terminal contained the output logs of Warp.

The Warp documentation doesn't include changing the integrated terminal and neither does the vscode documentation, so I highly suspect this is not feasible, but I was wondering if there was some less obvious workaround that someone might have found.