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

On a Conjecture of Lan-Sheng-Zuo on Semistable Higgs Bundles: Rank 3 Case