Non-vanishing conjecture for Bernoulli numbers modulo primes

Let $k\geq 3$ be an odd integer. The Bernoulli number $B_{p-k}$ is defined for each prime $p$.

Non-vanishing conjecture. There exist infinitely many primes $p$ such that

$$B_{p-k}\not\equiv 0\pmod{p}.$$

This is equivalent to the non-triviality of the corresponding element of the algebra of finite multiple zeta values. The statement is presented as an open problem in the source, and no resolution is supplied here.