A complex orthogonal design (COD) with parameter [p, n, k] is a combinatorial design used in space-time block codes (STBCs). For STBCs, n is the number of antennas, k/p is the rate, and p is the decoding delay. A class of rate 1/2 CODs called balanced complex orthogonal designs (BCODs) has been proposed by Adams et al., who constructed BCODs with rate k/p = 1/2 and decoding delay p = 2mfor n = 2m. Furthermore, they proved that the constructions have optimal decoding delay when m is congruent to 1, 2, or 3 modulo 4. They conjectured that for the case m ≡ 0 (mod 4), 2mis also a lower bound on p. In this paper, we prove this conjecture.