Vanishing conjecture for the Bloch–Kato term $h_{\mathrm{BK}}$

Let $X$ be the smooth projective curve associated with the notation in the paper, let $V_p\operatorname{Jac}_X$ be the $p$-adic representation attached to its Jacobian, and define

$$h_{\mathrm{BK}}\coloneqq\dim_{\mathbb Q_p}H^1_f\left(G_{\mathbb Q},\operatorname{Hom}\left(\bigwedge^2V_p\operatorname{Jac}_X,\mathbb Q_p(1)\right)\right).$$

Bloch–Kato vanishing conjecture. One expects

$$h_{\mathrm{BK}}=0.$$

The paper uses this as a conjectural consequence of the Bloch–Kato conjectures. Its vanishing would remove a term that currently obstructs weaker hypotheses for the finiteness and explicit bounds obtained from the weight $\geq -2$ quotient.