Research surveys from the Houses of Parliament conclude that expanding grammar schools would both widen and cement inequality, far from shortening the gap. This is because more affluent children have in-built advantages when it comes to passing the 11+ (more access to a tutor, more socioeconomic capital), and going to a grammar tends to lead to higher salaries later in life which in fact makes the problem of inequality worse, not better.
The research on this is dead against you, good sir. This is before of course we examine the teaching crisis in England which means comprehensives are already well short of the staff they need. Building more grammars will see more teachers simply migrate to them and leave comps in an even worse state.