Safari Quick Website Search | How to tell Safari the way my full-text search works?

The Apple browser Safari has a feature called "Quick Website Search". If you write the name of a website that you have already visited in the address bar, you can search directly on the page with a space followed by a search word, e.g. "widipedia dog".

On my Typo3 website I use a Solr full-text search with auto-complete. Safari recognizes these and queries them when using the Quick Website Search. The call Safari uses looks like this:

https://www.example.com/index.php?id=0&L=0&eID=tx_solr_suggest&format=OpenSearch&q=<search-term>

It's very smart, that Safari automatically detects the search-engine, but unfortunately it's wrong. The page ID is missing. It should look like this:

https://www.example.com/index.php?id=1234&L=0&eID=tx_solr_suggest&format=OpenSearch&q=<search-term>

Even if you confirm the search, the wrong page will be displayed:

https://www.example.com/index.php?id=0&L=0&q=<search-term>

It should look like this:

https://www.example.com/index.php?id=1234&L=0&q=<search-term>

Or even better:

https://www.example.com/search/?q=<search-term>

Does anyone know how I can tell Safari what the search query should look like?

Please excuse my english. Many thanks and best regards.