We note that the profit from theta-weighted variance dispersion is roughly the difference between implied and realized correlation multiplied by the average single-stock volatility. As correlation is correlated to volatility, this means the payout when correlation is high is increased (as volatility is high) and the payout when correlation is low is decreased (as volatility is low). A short correlation position from going long dispersion (short index variance, long single-stock variance) will suffer from this as profits are less than expected and losses are greater.
We note this does not necessarily mean a long dispersion trade should be profitable (as dispersion is short vol of vol, the fair price of implied correlation is above average realized correlation)
Dispersion is therefore short vol of vol; hence, implied correlation tends to trade c10 correlation points more than correlation swaps (which is c5 points above realized correlation).
The payout of a theta-weighted dispersion is therefore equal to the difference in implied and realised correlation (market value-weighted pairwise realised correlation) multiplied by the weighted average variance. If vol of vol was zero and volatility did not change, then the payout would be identical to a correlation swap and both should have the same correlation price. If volatility is assumed to be correlated to correlation (as it is, as both volatility and correlation increase in a downturn) and the correlation component is profitable, the profits are reduced (as it is multiplied by a lower volatility). Similarly, if the correlation suffers a loss, the losses are magnified (as it is multiplied by a higher volatility). Dispersion is therefore short volga (vol of vol) as the greater the change in volatility, the worse the payout. To compensate for this short volga position, the implied correlation level of dispersion is c10 correlation points above the level of correlation swaps.
I wrote a basic thread explaining correlation trading including how to estimate implied correlation, described how short corr is short vol-of-vol, and how to look for the presence of correlation more broadly. (Link)