The double marginalization problem makes the (monopoly) producers worse off as well as making consumers worse off.
Now as noted in the video combining the seller monopoly with the buyer monopoly into one will, in theory, make both the producers and consumers better off. But does it do so in practise? At least as far as consumers are concerned.
This is where a new paper Does Vertical Integration Decrease Prices? Evidence from the Paramount Antitrust Case of 1948 by Ricard Gil in the American Economic Journal: Economic Policy, 2015, 7(2): 162–191, helps us. Gil finds, utilising data from the Paramount antitrust case of 1948, that vertical integration, i.e. combining the two monopolies, does in fact decrease prices, and thus increases consumer welfare, in a manner consistent with the elimination of the double marginalization problem.
The paper's abstract reads:
I empirically examine the impact of the 1948 Paramount antitrust case on ticket prices using a unique dataset collected from Variety magazine issues between 1945 and 1955. With information on prices, revenues, and theater ownership for an unbalanced panel of 393 theaters in 26 cities, I find that vertically integrated theaters charged lower prices and sold more admission tickets than nonintegrated theaters. I also find that the rate at which prices increased in theaters was slower while integrated than after vertical divestiture. These findings together with institutional details are consistent with the prediction that vertical integration lowers prices through the elimination of double marginalization.