The code requirements for separation are more of a safety requirement, but must be at least 12" apart (note that this may vary with your local AHJ - Authority Having Jurisdiction). As a general rule, you want them in separate tray. Please see the NEC (NFPA 70) for specific requirements.
The idea is that you do not want someone working on low voltage (data cabling) to accidentally come in contact with high voltage (light you up like a Christmas tree) cabling. Interference is possible, but not as likely with higher grades of cabling provided that you obey the separation rules. If you follow the safety code, you are less likely to be in a situation where interference can occur as separation will be in place anyway. But they are correct to be concerned. You do not want them side by side in the same tray.
I am curious why you are not considering Augmented Category 6 that will also support 10GBASE-T applications which will be out in the very near future. You can support some distances on legacy category 6, but may have to do some mitigation to eliminate alien cross-talk. (Example: not bundling cables.)
When you install the cabling you will want to check out TIA TSB-155 so that you don't have to revisit the cabling when you move to 10GBASE-T in the future. No sense in adding back labor. You can check our website for podcasts and other articles that address the mitigation techniques. Once and done is my philosphy.
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