  A curve over a perfect field $K$ of characteristic $p > 0$ is said to be
superspecial if its Jacobian is isomorphic to a product of supersingular
elliptic curves over the algebraic closure $\overline{K}$. In recent years,
isomorphism classes of superspecial nonhyperelliptic curves of genus $4$ over
finite fields in small characteristic have been enumerated. In particular, the
non-existence of superspecial curves of genus $4$ in characteristic $p = 7$ was
proved. In this note, we give an elementary proof of the existence of
superspecial nonhyperelliptic curves of genus $4$ for infinitely many primes
$p$. Specifically, we prove that the variety $C_p : x^3+y^3+w^3= 2 y w + z^2 =
0$ in the projective $3$-space with $p > 2$ is a superspecial curve of genus
$4$ if and only if $p \equiv 2 \pmod{3}$. Our computational results show that
$C_p$ with $p \equiv 2 \pmod 3$ are maximal curves over $\mathbb{F}_{p^2}$ for
all $3 \leq p \leq 269$.
