This paper is devoted to the $L^2$ norm error estimate of the Adini element for the biharmonic equation. Surprisingly, a lower bound is established which proves that the $L^2$ norm convergence rate cannot be higher than that in the energy norm. This proves the conjecture of [Lascaux and Lesaint, RAIRO Anal. Numer., 9 (1975), pp. 9--53] that the convergence rates in both $L^2$ and $H^1$ norms cannot be higher than that in the energy norm for this element.