Equipment was not provided by the state (Apart from legionaires, not elites anyways) so you might as well replace that with wealth, instead of just equipment, as well as a multitude of factors.
Furthermore, you argument doesn't apply to your theory. The mere fact that Liby-Phoenican infantry are fighting in a unit outside of the Sacred Band makes them ineligible as Carthaginian militia don't turn into Libyans.
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