Thangadurai's conjectural refinement of Adams's theorem for Bernoulli numerators

Let $p>3$ be a prime, let $n$ be a positive even integer, and write $p^l\mathrel{\|}n$ to mean that $p^l\mid n$ but $p^{l+1}\nmid n$. Assume that $p-1\nmid n$. Write $p^\beta\mathrel{\|}N(B_n)$ to mean that $p^\beta$ exactly divides the numerator $N(B_n)$ of the Bernoulli number $B_n$. Thangadurai's conjecture. If

$$p^l\mathrel{\|}n,$$

then

$$p^\beta\mathrel{\|}N(B_n)\quad\Longrightarrow\quad \beta\leq l+1.$$

This is presented as a more recent conjectural version of Adams's theorem, which gives a related divisibility result for Bernoulli numbers. The source gives no resolution status.