Is the cotangent bundle to $P^2$ a toric variety?

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Is the cotangent bundle to the complex projective plane, $T^* P^2$, a toric variety? More generally, is there a criterion for when a vector bundle over a toric variety is a toric variety?

For example, in this related question When are vector bundles on toric varieties also toric varieties? it is suggested that if it splits, then it is a toric variety.