Well salty, you can give it a go, if it works, then be happy. It's hardly going to leave you out of pocket, trying this method(bearing in mind reaped application of toothpaste and lint free very soft cloth (probably best to use microfibre one, or microclear should you desire)),
It won't work, bro.
MOHS's hardness level of toothpaste (varies, but they won't get anywhere near sapphire crystal level): Toothpaste Abrasivity