If the person posting was prepared to attend a tribunal hearing in the full knowledge that he/she could prove the precise amount of their actual income (assuming it is below the minimum wage) then I would hope that DDC would apply the relevant discount. That would probably be a more effective approach than using social media.
That said, failure to disclose property as a means of avoiding council tax is fraud. In Cumming's case, it seems that his council regard that as permissible.
