Conjecture on the independence number of the fourth power of the 5-wheel

Let $W_5$ be the join of the cycle $C_5$ with $K_1$, let $\alpha(G)$ denote the independence number of $G$, and let $W_5^{\Box 4}$ be the fourth Cartesian power of $W_5$. Fourth-power independence-number conjecture.

$$\alpha(W_5^{\Box 4})=338.$$

Heuristic computer searches provide the lower bound $\alpha(W_5^{\Box 4})\geq 338$, and the conjectured equality would improve the bound for $\mathscr{I}(W_5)$; the equality remains unresolved.