This is true. Insurance companies employ people like me to inspect cars after a collision. The first thing I check is tyres. Not just tread depth, but speed and load rating too.
Incorrect load index will fail an MOT (a keen tester). My LCR has a 91 XL load rating. Its not just static weight the tyre needs to support, but also cornering stress. Bottom line...dont risk it. Dont give your insurance an excuse to not to pay out.