Conjectural Bernoulli recursion for coefficients of the Artin–Hasse exponential

Let $p$ be an odd prime and let

$$\operatorname{E}_{p}(X)=\sum_{i=0}^{\infty}a_iX^i$$

be the Artin–Hasse exponential series in $\mathbb F_p[[X]]$. For an integer $n$, let $B_n$ denote the $n$-th Bernoulli number. The conjectural recursion. For every integer $k$ with $1<k<p$, one has

$$\sum_{r=0}^{k}(-1)^r r a_{rp}a_{(k-r)p}=\frac{B_{p-k}}{k}.$$

The formula would provide a replacement for the preceding recursion, whose useful conclusion is restricted to even indices, and would allow the coefficients $a_{kp}$ with odd $k$ to be obtained recursively. The required higher supercongruences for direct calculations are currently unavailable, so the assertion is presented as conjectural.