Closed-form conjecture for the near-extremal asymptotic coefficients

Let $k\geqslant4$, and let $a_{k,j}$ be the coefficients in the asymptotic expansion, with parameters $\alpha$, $\varkappa$, $\varepsilon$, $b$, and $a_{1,\pm1}$ as defined in the paper. Near-extremal coefficient conjecture. For $k\geqslant4$,

$$a_{k,\pm(k-2)}=\pm\frac{\mathrm{i}\sqrt{3}\,a_{1,\pm1}^{k-2}}{2^k3^k(\varepsilon b)^{1/3}}\left(3(k-2)^2(\alpha^2+8\varkappa\alpha+10\varkappa^2)-5(k-2)^2+24(k-2)-24\mp12(k-2)^2\alpha\mp6\big(5(k-2)^2+8(k-2)\big)\varkappa\right).$$

The formula is extrapolated from explicit computations at levels through $k=10$; the supplied text does not establish it for all $k\geqslant4$.