I call attention to the Inverse-Hirschman scenario's twist for a specific historical reason. It basically describes another historical situation--this one in Charlottesville, Virginia in the spring of 1959, amidst ongoing court orders to desegregate.
The Charlottesville school system's attorney at the time was John S. Battle, Jr.--a committed segregationist politician and son of a former Virginia governor with close ties to the political machine of Senator Harry Flood Byrd, Sr. Virginia's legislature was debating the adoption of a tuition grant program that had some characteristics of a rudimentary voucher system. Although ostensibly race-neutral, these tuition grants could be used to send white children to "segregation academies" in place of the public school. This proposal is what modern voucher opponents use to link vouchers and segregation.
Not all segregationists supported the tuition grants, though, and one of their vocal opponents was none other than John S. Battle, Jr. Speaking at a Charlottesville PTA meeting on March 23, 1959, Battle denounced the program and implored white parents to exercise solidarity by registering their children in the public schools. His rationale was simple. By keeping segregationist parents around as a social buffer against integration, the white-controlled public school system could limit the admission of black students to only a few families, despite the court orders to integrate. They would do so by keeping white-majority schools at constant full capacity, leaving no space for more black students.