Irregular-prime characterization for a half-power-sum congruence

Let $S_m(n)=\sum_{j=1}^{n-1}j^m$, and let $B_m$ denote the $m$th Bernoulli number. An irregular prime is a prime dividing the numerator of some Bernoulli number $B_{2r}$ with $2\le 2r\le p-3$. Half-power-sum characterization conjecture. For an odd positive integer $n\ge 3$ with $n\equiv 3\pmod{4}$, the congruence

$$S_{(n-1)/2}(n)\equiv 0\pmod{n^3}$$

holds if and only if $n$ is an irregular prime for which

$$B_{(n-3)/2}\equiv 0\pmod{n}.$$

The claim is motivated by congruences relating power sums to Bernoulli numbers and computations through $n=20000$; its status is unresolved in the supplied text.