jq select error: "Cannot index string with string <object>" - json

Question

command:
cat test.json | jq -r '.[] | select(.input[] | .["$link"] | contains("randomtext1")) | .id'
I was expecting to have both entries (a and b) to show up since they both contains randomtext1
Instead, I got the following output message:
a
jq: error (at <stdin>:22): Cannot index string with string "$link"
From some digging I understand that the issue is likely caused by the following object/value pair in the a entry:
"someotherobj": "123"
because it does not contain the object $link and the filter in the command expects to see $link in all objects under the input so it errors out before the command has a chance to search in the b entry.
What I really want is to be able to search for any entries that have at least one "$link": "randomtext1" pair under input. Is there a fuzzier search feature allowing me to achieve this?
I tried to use two contains hoping it will just pipe things through:
jq -r '.[] | select(.input[] | contains(["$link"]) | contains("randomtext1")) | .id'
but it did not like that at all..
the test.json file:
[
{
"input": {
"obj1": {
"$link": "randomtext1"
},
"obj2": {
"$link": "randomtext2"
},
"someotherobj": "123"
},
"id": "a"
},
{
"input": {
"obj3": {
"$link": "randomtext1"
},
"obj4": {
"$link": "randomtext2"
}
},
"id": "b"
}
]

Answer 1

What I really want is to be able to search for any entries that have at least one "$link": "randomtext1" pair under input.
The key word here, both in the question and the following answer, is any:
.[]
| select( any(.input[];
type=="object" and has("$link") and (.["$link"] | index("randomtext1"))))
| .id
Of course if you require the key's value to be "randomtext1", you'd write .["$link"] == "randomtext1".