Packing version of Dinitz's problem

For an integer $n\geq 3$, let $K_n$ be the complete graph on $n$ vertices, let $K_n\mathbin\square K_n$ denote their Cartesian product, and let $\chi_\ell^\star$ denote the list packing number.

Packing version of Dinitz's problem. For $n\geq 3$,

$$\chi_\ell^\star(K_n\mathbin\square K_n)=n+1.$$

The graph $K_n\mathbin\square K_n$ is the line graph of $K_{n,n}$, and its ordinary list chromatic number is $n$ by Galvin's theorem, whereas the paper shows that its list packing number exceeds $n$. Determining whether the exact value is $n+1$ remains open.