Please explain if it is proven that loose shoes cause clawed toes?

Not sure if there. Is such a study. Genetics plays a role as does imbalance of tendons. If shoes are too big the toes may flex early to try and grab the shoe...So intuitively it may contribute.
Not physiologic. No direct relation between claw toes and loose shoes. Claw toes are caused by chronic overuse of the toe extensors to aid in ankle dorsiflexion. This results in anterior leg compartment muscle hypertrophy, which results in the chronic over pull of these tendons. As a result, drawing up of the toes, or clawing occurs.
May contribute. Not proven, but it may be a contributing factor due to tendon overuse.