There are many kinds of footwear that people will argue is best from sneakers to steel arched hiking boots.
Heels? Nope, says so on the sign.
Now I bet your thinking that ain't right, you(they) should be able to wear what you want.
But if you do you'll probably get hurt maybe break an ankle. Then you'll be looking for paramedics, a mountain rescue team, an ambulance, maybe even a helicopter. All that manpower(money) and resources(mo money) to save your dumb ass.
Not to mention the cost of insurance to cover, and lawyers to defend the trail owner, medics, rescue team and shoe manufacturer from your lawsuits.