Parity and extremal-coefficient conjecture for the asymptotic expansion

Let $a_{k,j}$ be the coefficients in the asymptotic expansion indexed by $k\in\mathbb{N}$ and $j\in\mathbb{Z}$ with $|j|\leqslant k$. Parity and extremal-coefficient conjecture. If $k$ and $j$ have different parity, then

$$a_{k,j}=0,$$

and

$$a_{k,\pm k}=\frac{k\,a_{1,\pm1}^k}{2^{k-1}3^{k-1}}.$$

The claim is inferred from explicit calculations through level $k=10$ and is proposed as a general pattern; the supplied text does not provide a proof or resolution.