This is simply untrue. Just take this simple example, suppose you've entered a position ATMF. Gamma is going to be the same for puts/call, let's say gamma = g. Then the gamma of a straddle will be 2*g (or -2*g for short). In a wrangle you're long both ratio spreads for calls and puts, so let's say short both ATMF calls and puts which will put you at -2*g gamma, however you've purchased 2x in the wings which will bring your gamma down close to zero.
Another way to look at the wrangle is the cost of trading vega convexity.