Final answer:
The historical challenge faced by Zeta Phi Beta Sorority, Inc. in attracting members at Howard University might be attributed to existing competition and broader racial integration challenges.
Step-by-step explanation:
The major reason that Zeta Phi Beta Sorority, Incorporated lost many of its prospective members in the beginning on Howard University's campus is not explicitly detailed in the provided information. However, we can glean from the historical context that challenges related to racial integration and the competition among established sororities may have played a role. For example, other sororities such as Alpha Kappa Alpha Inc., founded earlier at Howard University, had already established prominent membership and may have drawn potential members away. Furthermore, the broader societal issues of integration and racial tension at colleges, as noted within the historical context of other African-American institutions, might have influenced the sorority's initial membership numbers.