Toric polynomial generator conjecture for complex cobordism

Let $\Omega_{*}^{U}$ be the complex cobordism ring, with polynomial generators

$$\Omega_{*}^{U}\cong\mathbb{Z}[\alpha_{1},\alpha_{2},\ldots].$$

For each $n\geq 1$, a smooth projective toric variety should have a cobordism class that can be chosen as the polynomial generator $\alpha_n$.

Toric polynomial generator conjecture. For each $n\geq 1$, there exists a smooth projective toric variety whose cobordism class can be chosen for the polynomial generator $\alpha_n$ of $\Omega_{*}^{U}$.

Smooth projective toric varieties are simultaneously connected, algebraic, and combinatorially tractable, so this conjecture would provide convenient polynomial generators for complex cobordism. The source reports constructions in all odd dimensions and in dimensions one less than a prime power, while the remaining even dimensions are supported by numerical evidence but are not established.