Can the REVIAN cap be folded for travel?

The cap isn't hard plastic like some competitors products but is cannot be folded. The REVIAN cap is predominantly fabric and foam but it has a flexible inner lenticular lens that will crack or break if folded.

The REVIAN cap is super lightweight. When I travel I pack clothes in and around the cap and it takes up hardly any space. It will easily fit in standard size carry-on luggage.