Néron–Severi rank and exterior-square L-function order

Prove that for every smooth projective curve X over the rationals, the rank of the Néron–Severi group of its Jacobian equals the negative of the order at s=2 of the exterior-square L-function L(∧²X,s).

Background

This conjecture is presented as a special case of Tate’s conjecture relating algebraic cycles to orders of poles of L-functions. In the paper’s framework, it follows by combining the conjectured residual-motive nonvanishing with the established identification of the Sato–Tate moment M₁[a₂] and the Néron–Severi rank.