S3 bucket policy: In a Public Bucket, make a sub-folder private

I have a bucket filled with contents that need to be mostly public. However, there is one folder (aka "prefix") that should only be accessible by an authenticated IAM user.

{
  "Statement": [
    {
      "Sid": "AllowIAMUser",
      "Action": [
        "s3:GetObject"
      ],
      "Effect": "Allow",
      "Resource": "arn:aws:s3:::bucket/prefix1/prefix2/private/*",
      "Principal": {
        "AWS": [
          "arn:aws:iam::123456789012:user/bobbydroptables"
        ]
      }
    },
    {
      "Sid": "AllowAccessToAllExceptPrivate",
      "Action": [
        "s3:GetObject",
        "s3:GetObjectVersion"
      ],
      "Effect": "Allow",
      "Resource": "arn:aws:s3:::bucket/*",
      "Condition": {
        "StringNotLike": {
          "s3:prefix": "prefix1/prefix2/private/"
        }
      },
      "Principal": {
        "AWS": [
          "*"
        ]
      }
    }
  ]
}

When I try to save this policy I get the following error messages from AWS:

Conditions do not apply to combination of actions and resources in statement -
  Condition "s3:prefix"
  and action "s3:GetObject"
  in statement "AllowAccessToAllExceptPrivate"

Obviously this error applies specifically to the second statement. Is it not possible to use the "s3:prefix" condition with the "s3:GetObject" action?

Is it possible to take one portion of a public bucket and make it accessible only to authenticated users?

In case it matters, this bucket will only be accessed read-only via api.

This question is similar to Amazon S3 bucket policy for public restrictions only, except I am trying to solve the problem by taking a different approach.