A maximal $T$-space of $\mathbb{F}_{3}[x]_0$

In earlier work, we have established that for any finite field $k$, the free associative $k$-algebra on one generator $x$, denoted by $k[x]_0$, has infinitely many maximal $T$-spaces, but exactly two maximal $T$-ideals (each of which is a maximal $T$-space). However, aside from these two $T$-ideals, no specific examples of maximal $T$-spaces of $k[x]_0$ were determined at that time. In a subsequent work, we proposed that for a finite field $k$ of characteristic $p>2$ and order $q$, for each positive integer $n$ which is a power of 2, the $T$-space $W_n$, generated by $\{x+x^{q^n}, x^{q^n+1}\}$, is maximal, and we proved that $W_1$ is maximal. In this note, we prove that for $q=p=3$, $W_2$ is maximal.