Coprocess FD not available when duplicated to FD other than 1 or 2

[lolilolicon]

The following correctly prints "fd=2":

    coproc p { head -1; }
    2>&${p[1]} bash -c 'echo fd=2 >&2'
    cat <&${p[0]}

Change 2 to 3, throws error: "bash: 3: Bad file descriptor":

    coproc p { head -1; }
    3>&${p[1]} bash -c 'echo fd=3 >&3'
    cat <&${p[0]}

I'm not very familiar with redirection, so I may have overlooked
something obvious...

What am I missing?