Actually, no. What is the cone made of? LDPE most likely. Soft, easily melted, easily shredded, easily burned.
So a gas turbine most likely won't even notice a cone.
Besides in a modern high-bypass turbofan about 80% of it would just get shredded by the fan and fly around the delicate parts, which aren't that much delicate.
What the cone is made out of is irrelevant. Any sort of FOD ingestion will lead to downtime and inspection/rebuild depending on severity of the damage found.
“One of these 15 million dollar jet engines, just being disabled by a common traffic cone”
I can think of a lot of really expensive equipment that can be destroyed or disabled by a common traffic cone.