Consider a saturated main effect plan for the parametric vector $\beta_1$, then this paper shows that when this plan is optimum (in the sense of maximum determinant of the design matrix) then the complementary plan is also optimum for the complementary vector $\beta_2$. Some additional corollaries are derived for special cases.
B. L. Raktoe. W. T. Federer. "A Theorem on Saturated Plans and Their Complements." Ann. Math. Statist. 41 (6) 2184 - 2185, December, 1970. https://doi.org/10.1214/aoms/1177696728