Replacing a JSON array in MySQL in PHP

Question

I am trying to replace data in a JSON array inside a JSON array in my MySQL Database.

The array looks like this:

{"slug": "SLUG","price": "{"44":12,"_default":12}", "test": "TEST"}

Now I would like to update the two fields inside the price array.

I am using the following code to be able to replace the price array with a string or number, but I am not able to replace it with another array:

$sql = "UPDATE products SET productDynamicFields = JSON_REPLACE(productDynamicFields,'$.price', '$test')
        Where productSlug = '$productSlug'";
$result = mysqli_query($link,$sql);

As you can see I am using a variable called $test.

I tried giving the variable the following value: $test = ['44' => 13, '_default' => 13]; and then encoding it like this: $test = json_encode($test);

But now in my database it looks like this:

"price": "{\"44\":13,\"_default\":13}

I now tried using JSON_UNESCAPED_SLASHES with no success.

What can I do to achieve the array inside the array?