Linear product-of-characters conjecture for classical and alternating groups

For characters $\chi_1,\ldots,\chi_l$ of $G$, write

$$\chi_1\chi_2\cdots\chi_l\supseteq\operatorname{Irr}(G)$$

when every irreducible character of $G$ is a constituent of their product. Linear product-of-characters conjecture. There is an absolute constant $C>0$ such that, for $G=\operatorname{Cl}_n(q)$ a classical simple group of dimension $n$, or $G=\mathsf{A}_n$ an alternating group of degree $n\geq5$, whenever $l\geq Cn$ and $\chi_1,\ldots,\chi_l\in\operatorname{Irr}(G)\setminus\{1_G\}$, one has

$$\chi_1\chi_2\cdots\chi_l\supseteq\operatorname{Irr}(G).$$

This is proposed as a more modest linear-length analogue of the product-covering result. It remains open in the source.