Improving certain under performing middle schools in poorer neighborhoods, by turning them into charter schools and giving vouchers to parents in those districts, will do massively more to improving the demographics of who enters the elite public high schools, than any bureaucratic policy trying to merely dictate forced changes in those demographics.
The current demographics are 100% NOT about the exams to enter the elite public schools. They are 100% about which groups of students do not obtain the education they need BEFORE high school that would put them in better stead to pass those exams. Some % of that is about (a) the middle schools they attend, and some of it is about (b) their family & home life and the degree of their own academic dedication. The split between the two would vary with each student.
The biggest impediment to educational progress is the fantasy of equal potential. (The fantasy has the effect of hurting just about everyone, as it interferes with the recognition of actual needs and potentials.)