Many economic commenters, when broaching into the topic of induced demand, like to break into a car analogy. They assert more lanes actually cause more traffic rather than less.

Is there any proof of that assertion?

The few studies I've found on this suggest otherwise.

That is, for an expansion of x% capacity, we get induced demand of y%, where y is less than x (sometimes by several factors).

Which suggests adding lanes can help reduce congestion. 

On the network itself, before we even look at decreased congestion on other means of transportation caused by this induced car demand.

Haven't looked at any research data tbh, but guessing it just has to depend on more than that one factor? Location (mainly population density of the area), quality and availability of public transportation, parking availability in the (broader) area?