Tu's elliptic expansion conjecture for the modular j-function

Let $s(w)=2\pi\Omega^2S(w)$ be the rescaled inverse mirror-map coordinate, and let $g(t)$ and $h(t)$ be the power series associated with the elliptic quartic as defined above. Around the hexagonal point and the square point, use the corresponding local expansions of the modular $j$-function. Elliptic expansion conjecture. Around the hexagonal point,

$$j\left(s^{-1}\left(\frac{h(t)}{g(t)}\right)\right)=27t^3\left(\frac{8-t^3}{1+t^3}\right)^3 =13824t^3-46656t^6+99144t^9-171315t^{12}+263169t^{15}-\cdots,$$

and around the square point,

$$j\left(s^{-1}\left(\frac{h(t)}{g(t)}\right)\right)=(192+256t)\left(\frac{3+4t}{1-4t^2}\right)^2 =1728+20736t^2+147456t^4+851968t^6+4456448t^8+\cdots.$$

These identities would relate the categorical Saito-theoretic flat coordinates for the elliptic quartic to the classical modular parameter, extending the analogous description for Fermat cubics; the source provides no resolution, so the conjecture remains open.