Determine the list strong rainbow connection number of wheels

Determine the list strong rainbow connection number $src^\ell(W_n)$ of the wheel graph $W_n$ for every integer $n\ge 3.

Background

The paper establishes exact values of $src^\ell(W_n)$ for $3\le n\le 9$ and derives lower and upper bounds for $n\ge 10$. It also proves that, for $n\ge 7$, determining $src^\ell(W_n)$ is equivalent to determining the list chromatic number of the complement of the square of the cycle, $\chi_\ell(\overline{C_n^2})$. The authors state that the correct value is expected to be substantially smaller than the available upper bound, leaving the exact determination unresolved.