@dannythomas I think Stanford fully surpassed Yale and Princeton in terms of academic/research prestige and started being widely recognized as such at least ~10 years ago. (Look at yields, rankings of departments and international rankings which actually measure strength in specific fields, fundraising, among other metrics, the various articles debating whether Stanford has surpassed Harvard as the nations top school etc etc).
The main rivalry now is between Harvard and Stanford for top spot, but pretty sure Harvard still wins in terms of lay prestige (i.e. among average, not-so-well-educated people).
Of course this is beyond splitting hairs and anyone would be insanely lucky to get into any of HYPS.