I do not agree with the statement that volswap is short convexity or that varswap is long convexity etc. It depends what your 'base' instrument is: if it is the varswap then the volswap has convexity (actually concavity), if it is the volswap then the varswap has convexity. I.e. convexity is a relative measure. However given the existence of both the varswap and the volswap, then their difference is a measure of convexity or vol of vol, and since the volswap is then the linear instrument the varswap is long convexity, which means the varswapvolswap is then "short" convexity / vol-of-vol. But again, it is a relative measure.