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We study the Picard rank of smooth toric Fano varieties possessing familiesof minimal rational curves of given degree. We discuss variants of a conjectureof Chen-Fu-Hwang and prove a version of their statement that recovers theoriginal conjecture in sufficiently high dimension. We also prove new cases ofthe original conjecture for high degrees in all dimensions. Our main tools comefrom toric Mori theory and the combinatorics of Fano polytopes.

Bounds on the Picard rank of toric Fano varieties with minimal curve constraints