Nonvanishing conjecture for Bernoulli-number finite zeta values

For every odd integer $k\geq 3$, define the element $(B_{p-k}\pmod p)_p\in\mathcal{A}_{\mathbb{Q}}$, where $B_m$ denotes the $m$-th Bernoulli number. Bernoulli finite zeta nonvanishing conjecture. For all odd integers $k\geq 3$,

$$(B_{p-k}\pmod p)_p\in\mathcal{A}_{\mathbb{Q}}$$

is non-zero. This is naturally related to the Kaneko–Zagier conjecture and predicts nonvanishing of a family of finite multiple zeta values. The source provides no resolution status for this conjecture.