No, the UTXO blocks, the ones that matter for the network, don't have signature data. The witness blocks are not a part of UTXO and therefore are optional. If they are ommited there is no consequence. And it would be more profitable to ommit them.
If you're running an honest and upgraded segwit node, your node will recognise the witness and verify everything in it. Given that most of the network nodes run an upgraded node that is aware, the nodes will automatically reject those blocks with mismatching signatures. You can omit it if you're a miner, you will just be on a fork which the majority of the network isn't following.
If no one is following it, your chain is worthless.