Let $u$ be a biharmonic Green potential on the unit ball $\mathbf{B}$ of $\mathbf{R}^{n}$.
We show that
\begin{equation*}
\lim_{r\to 1}(1-r)^{n-2-(n-1)/p}\mathcal{M}_p(u,r)=0
\end{equation*}
for $p$ such that $1\le p<(n-1)/(n-4)$ in case $n\ge 5$
and $1\le p<\infty$ in case $n\le 4$.
Further, if $n\ge 5$ and $(n-1)/(n-4)\le p<(n-1)/(n-5)$,
then it is shown that
\begin{equation*}
\liminf_{r\to 1}(1-r)^{n-2-(n-1)/p}\mathcal{M}_p(u,r)=0.
\end{equation*}
Finally we show that these limits characterize biharmonic Green potentials
among super-biharmonic functions on $\mathbf{B}$.