Conjecture on the distribution of irregular pairs by index difference

Let $M$ be a fixed positive integer and let $c$ be an integer with $0\le c<M$. An irregular pair is a pair $(p,t)$ such that $p$ is prime, $t$ is even, and $p$ divides the Bernoulli number $B_t$. The irregular index of a prime $p$ is the number of irregular pairs with first component $p$. Distribution conjecture. We have

$$\lim_{X\to\infty}\frac{\#\{(p,t): p\mid B_t,\ t\text{ even},\ p<X,\ p-t\equiv c\pmod M\}}{\#\{(p,t): p\mid B_t,\ t\text{ even},\ p<X\}}= \begin{cases} 0 & \text{if $2\mid c$ and $2\mid M$,}\\ 1/M & \text{if $2\nmid M$,}\\ 2/M & \text{if $2\nmid c$ and $2\mid M$. }\end{cases}$$

The same assertion should hold after restricting to irregular primes with a fixed irregular index. This predicts an even distribution of irregular pairs according to $p-t$ modulo $M$, subject only to the parity obstruction when $M$ is even.