Statistics → Access Log Filename → Any path notation & symlink fails

WordPress → W3 Total Cache (Pro version!) → General Settings → Statistics

Cache usage statistics: Enabled
Slot time: 60 secs
Slots collected: 60
Webserver: Apache. That’s indeed my webserver.
Access Log Filename: Pressing TEST results in “Failed to open file” regardless what path I try.

1) My WordPress root directory is at: ~/html/apps/domain-x/ or as an absolute path /data/web/customer-ID/html/apps/domain-x/ .

2) ~/log/access.log is the real Apache access.log for my customer account (mixing all domains together, but I anyhow use only one production and one staging subdomain on that webhost for now). In SSH I can read access.log with cat or less.

3) According to my webhost PHP can only access until ~/html/ and not higher.

4) php_openbase is OFF to not further complicate things.

5) Regardless whether using an absolute path /data/web/customer-ID/ or a relative path, either relative to my home directory (starting with the ~ tilde) or relative to the WordPress root at ~/html/apps/domain-x/ (starting the path without a slash and without a tilde, so simply e.g. wp-config.php) nothing of the following worked:

~/log/access.log
/data/web/customer-ID/log/access.log

6) Hence I created two symlinks:
ln -s ~/log/access.log ~/html/all-my-domains-access.log
ln -s ~/log/access.log ~/html/apps/domain-x/all-my-domains-access-in-wp-root.log

Still nothing of the following worked:

~/html/all-my-domains-access.log
/data/web/customer-ID/html/all-my-domains-access.log
~/html/apps/domain-x/all-my-domains-access-in-wp-root.log
/data/web/customer-ID/html/apps/domain-x/all-my-domains-access-in-wp-root.log

7) I checked ~/log/php_error.log and found no time related log messages to when I click that TEST button. This test button failure leaves no trace in the logs. So I have no clue why it fails.

Help appreciated!