Possible Clues:

• Sounds cold, and if it's cold you'll need this
• A fine paper under consideration
• Though thin, it'll do for 4 down and sound all the same
• Being discussed
• Disputed a lot of lies?
• A matter of biological significance under discussion
• The matter being discussed is a thin, cold sound
• Under discussion, what lady keeps in her handbag?
• Sneeze said to be in dispute?
• Being fought over