jq: adding nested object to a JSON

I have a json file containing several profiles:

{"profile1": {"user": "user1","channel": "channel1","hook": "hook1"  },"default": {"user": "user1","channel": "channel1","hook": "hook2"  }}

I want to use jq to insert another profile, "test", so that the end result will be

{"profile1": {"user": "user1","channel": "channel1","hook": "hook1"  },"default": {"user": "user1","channel": "channel1","hook": "hook2"  },"test": {"user": "user3","channel": "channel3","hook": "hook3"  }}

Directly in the command line I can do it with:

cat filename |     jq  '.+  { "test": { "user": "u3", "channel" : "c3", "hook": "w3" } }'

But when I try it in my bash script:

cat "$CONF_FILE" |     jq --arg p "$PROFILE" --arg u "$U" --arg c "$C" --arg w "$WH" \'.+ { $p: { "user": $u, "channel": $c, "hook": $w } }' `

I'm getting the following error:

jq: error: syntax error, unexpected ':', expecting '}' (Unix shell quoting issues?) at <top-level>, line 1:.+ { $p: { "user": $u, "channel": $c, "hook": $w } }jq: error: syntax error, unexpected '}', expecting $end (Unix shell quoting issues?) at <top-level>, line 1:.+ { $p: { "user": $u, "channel": $c, "hook": $w } }jq: 2 compile errors

I've tried surrounding the variable with quotes but then I get just the string $p:

cat "$CONF_FILE" |     jq --arg p "$PROFILE" --arg u "$U" --arg c "$C" --arg w "$WH" \'.+ { "$p": { "user": $u, "channel": $c, "hook": $w } }'

result:

{"profile1": {"user": "user1","channel": "channel1","hook": "hook1"  },"default": {"user": "user1","channel": "channel1","hook": "hook2"  },"$p": {"user": "user3","channel": "channel3","hook": "hook3"  }}

EDIT: I have found what seems a temporary solution, converting the object to an array, editing the value (now the profile name is a value and not a key) and converting the array back to an object:

cat "$CONF_FILE" |     jq --arg p "$PROFILE" --arg u "$U" --arg c "$C" --arg w "$WH" \'to_entries | .+ [ { "key": $p, "value": { "user": $u, "channel": $c, "hook": $w } } ] | from_entries'

It seems crude, but it works. I am still hoping for a better solution...