The $\Delta$-conjecture for irregular primes and Bernoulli numbers

Let $p$ be an irregular prime. For an irregular pair $(p,l)$, let $\Delta_{(p,l)}$ denote the associated matrix, let $\Psi^{\rm irr}_1$ be the set of irregular pairs of order one, let $\Delta(p)$ be the corresponding structural invariant, and let $i(p),i_2(p),i_3(p),\ldots$ be the associated indices. A matrix is nonsingular when it has nonzero determinant.

$\Delta$-conjecture. The following properties are equivalent:

1. $\Delta_{(p,l)}$ is nonsingular for all irregular pairs $(p,l)\in\Psi^{\rm irr}_1$;
2. $\Delta(p)=1$;

$$i(p)=i_2(p)=i_3(p)=\ldots.$$

The conjecture asserts that these three descriptions are equivalent for every irregular prime, relating the nonsingularity of the matrices attached to irregular pairs to a structural property of the Bernoulli numbers and the stabilization of the associated indices. Its status is not established in the supplied source context.