  Let $X$ be a smooth projective curve of genus $g$ over an algebraically
closed field $k$ of characteristic $p>2$. We prove that any rank $3$ nilpotent
semistable Higgs bundle $(E,\theta)$ on $X$ is a strongly semistable Higgs
bundle. This gives a partially affirmative answer to a conjecture of
Lan-Sheng-Zuo \cite{LanShengZuo12ii}\footnotemark[1]. In addition, we prove a
tensor product theorem for strongly semistable Higgs bundles with $p$
satisfying some bounds (Theorem \ref{TensorTheorem}). From this we reprove a
tensor theorem for semistable Higgs bundles on the condition that the
Lan-Sheng-Zuo conjecture holds (Corollary \ref{TensorStableBundle}).
