How many queens are needed to attack all white squares?

This question: How can 3 queens control the white squares? got me thinking...

What is the fewest number of queens needed to attack every white square?
Rules:

• Only queens allowed
• Every white square is attacked — not just occupied
• No queen attacks another queen
• What do you mean by "attacked — not just occupied"? Every white square must be attacked, regardless of whether it's occupied or not? Or white squares can't be occupied? – Rand al'Thor Feb 8 at 19:39
• That was to clarify vs the linked question. In that question, it was enough for a queen to be on a white square (occupy). This question requires attacking all white squares. – Joel Rondeau Feb 8 at 19:41
• Correct me if I'm wrong (I probably am), but rule #2 and #3 makes it seem like queens can only be on black squares. All white squares must be attacked (not just occupied), meaning if a queen was on a white square, another queen would have to attack it, which breaks rule #3. If a queen is on a black square and we only care about attacking white squares, we only need to care about it's diagonal movement if it's attacking another queen, which would yield a possible solution like the naive symmetrical approach like this – Lukas Rotter Feb 8 at 19:47
• Your interpretation of rules #2 and #3 are correct. It seemed like explicitly stating that made the starting point too easy. – Joel Rondeau Feb 8 at 19:48
• @LukasRotter I was thinking that too ... but then I think there's no difference between queens and rooks, for the purposes of this problem? – Rand al'Thor Feb 8 at 19:52

4 queens: $$\begin{matrix}&.&.&.&.&.&.&.&.\\&.&.&.&.&Q&.&.&.\\&.&.&.&.&.&.&.&.\\&Q&.&.&.&.&.&.&.\\&.&.&.&.&.&.&.&.\\&.&.&.&.&.&.&Q&.\\&.&.&.&.&.&.&.&.\\&.&.&Q&.&.&.&.&.\\\end{matrix}$$