Analogy between product of conjugacy classes and irreps: is there analog of Thompson conjecture ?

In the following article

Heide, Gerhard; Saxl, Jan; Tiep, Pham Huu; Zalesski, Alexandre E. Conjugacy action, induced representations and the Steinberg square for simple groups of Lie type. Proc. Lond. Math. Soc. (3) 106 (2013), no. 4, 908–930.

Heide, Saxl, Tiep, and Zalesski show that if $G$ is a finite simple group of Lie type then every irreducible character is a constituent of $\mathrm{St}^2$ unless $G = \mathrm{SU}_n(q)$ where $n$ is coprime to $2(q+1)$, see Theorem 1.2. Here $\mathrm{St}$ denotes the Steinberg character of $G$.

In this exceptional case, namely $G = \mathrm{SU}_n(q)$ with $n$ coprime to $2(q+1)$, they show that there exists no irreducible character $\chi$ of $G$ such that either $\chi^2$ or $\chi\overline{\chi}$ contains all irreducible characters of $G$, see Lemma 5.3.

It's possibly worthwhile noting that the corresponding statement for the symmetric group is an open problem. If $n$ is a triangular number then Saxl has given a specific character of $\mathfrak{S}_n$ whose square conjecturally contains all irreducible characters; it's labelled by a staircase partition.

Remark: One can find a preprint version of the above article here.

As Taylor's answer shows, the simple group PSU(3,3) is a counterexample. Interestingly, the conjecture fails rather dramatically for this group since the irreducible character of degree 6 is not a constituent of $\chi\overline\chi$ for ANY irreducible character $\chi$ of $G$. This is a Magma computer observation, and I have no idea if a similar phenomenon occurs more generally.