It is extremely competitive and holistic in nature. Perfect scores and academic record are the starting point for consideration. Think tippy tops stats and scores.
Robertson looks for a track record of leadership and service that will thrive if given the support and exposure Robertson provides.
Not trying to dissuade you from trying but for context I am aware of a student who was a finalist for Robertson (ultimately denied) who was accepted at Duke, Harvard, Penn, Brown and Georgetown. Anecdotal but that leads me to believe that it is a harder acceptance than the low single digit acceptance rates in the RD round of the aforementioned schools.
It is however worth noting that the Robertson “experience” would for many make it as desirable as any and all collegiate opportunities. It is an incredible platform but extremely competitive and time consuming to successfully be considered.
This following was the case 5 years ago so pardon me if this has changed or my memory is off. Students who apply to UNC and Duke can be highly selectively put forward directly into Robertson and moved into semi final round. I believe this took place in February but again it has been some time. Those (think small number) have to quickly complete a variety of application essays and materials.
The entire semi finalist cohort (both referred by UNC/Duke admissions and direct applicants to Robertson) are then “rigorously” interviewed (rumor was around 150 kids but speculative). A short time later “finalists” are invited for final round interviews in NC (approximately 30-40 kids from each school). Winners are announced around Ivy day.
Hope this helps and good luck!!