  Suppose $f:S\rightarrow\mathbb{P}^1$ is a surface fibration of genus $g$ with
$3$ singular fibers and two of the fibers are semistable. In 1998, K. V. Nguyen
conjectured that such kind of fibration does not exist for $g\ge2$. But in
2013, C. Gong, X. Lu, and S.-L. Tan found a counterexample to Nguyen's
conjecture for $g=2$. Note that such kind of fibration shows strong arithmetic
properties, and as such the counterexamples are important, but rare in fact. In
this paper, a new counterexample to Nguyen's conjecture for $g=2$ is
constructed.
