Bernoulli--Seki nonvanishing consequence of regular-prime infinitude

For a non-negative integer $n$, let $B_n$ denote the $n$th Bernoulli--Seki number, defined by

$$\frac{te^t}{e^t-1}=\sum_{n=0}^{\infty}\frac{B_n}{n!}t^n.$$

Bernoulli--Seki nonvanishing consequence. Let $k\geq3$ be an odd integer. Then, there exist infinitely many primes $p$ greater than $k$ such that $p$ does not divide $B_{p-k}$. The paper states this as a consequence of the infinitude conjecture for regular primes, so it is open insofar as that conjecture remains open.