Unit-root subspace conjecture for canonical Mazur–Tate splittings

Let $F$ be a number field, let $v$ be a finite place above a prime $p$, and let $J/F$ be a Jacobian with semistable ordinary reduction at $v$. Let $\Theta$ be a theta divisor and set

$$X=\Theta+[-1]^*\Theta.$$

Assume that the canonical Mazur–Tate splitting is induced by a $p$-adic Néron function with respect to $X$ and a complementary subspace $W_v$. Unit-root subspace conjecture. The subspace $W_v$ is the unit root subspace of Frobenius. The source establishes this in the good ordinary reduction case covered by its genus-$2$ construction, but the assertion is conjectured in general whenever the preceding Néron-function conjecture holds.