Is anyone aware of the history of the following conjecture on the $p$-divisibility of (the numerators) of Bernoulli numbers?

CONJECTURE: For $p$ an odd prime, and $k$ even with $2 \leq k \leq p-3$, $p$ cannot divide both $B_k$ and $B_{p-k+1}$.

This conjecture is mentioned in Ohta's paper: Companion Forms and the structure of $p$-adic Hecke algebras. (see Example 3.3.4 on p. 25)

As far as I am aware, this is the only instance of the conjecture in literature. I believe I have a proof using companion forms and Ribet's proof of Herbrand-Ribet. I am currently writing it up and want to give proper references. Thanks