Unethical?

I have a coworker who is making all of our jobs harder but not doing his job. Everyone is able to see each other’s activity and our job is to take escalated inbound calls. Every single time he gets a call he immediately puts the customer on hold and on the surface it looks like he’s on a call but if you dig a little deeper you see that that ok is active for about a second and then he places them on hold. They eventually hang up and call back and this just repeats over and over. Management has yet to figure it out for 3-4 months as I said it looks like he’s on an inbound call. Would it be bad to fake an email and send him an email to scare him and let him know someone sees what he’s doing?

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