Bacher's conjecture on Bernoulli-number numerator quotients

Let $B_n$ be the Bernoulli numbers, let $(p,l)$ be an irregular pair with smallest $l$ when the index of irregularity $i(p)>1$, and let $\operatorname{num}(r)$ denote the numerator of a rational number $r$. Define

$$A(p)=\min_m\left\{m\,\middle|\,\operatorname{num}\left(\frac{B_m}{m}\right)\Big/\operatorname{num}\left(\frac{B_m}{m(m-1)}\right)=p\right\}.$$

Bacher's conjecture. Then $A(p)=(l-1)p+1$.

This conjecture concerns the numerators of the Bernoulli-number quotients $B_m/m$ and $B_m/(m(m-1))$ and was suggested by Roland Bacher in connection with Integer Sequence A092291. The paper later gives an equivalent formulation in terms of irregular pairs and proves a criterion for when the claimed value is valid; the conjectural assertion is not resolved here.