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Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2\pi}\frac{1}{\eta^{3/4}} \left[\sin \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)\right],\;\;\eta\gg 1.$$

see plot, blue is $F(\eta)$, gold is the large-$\eta$ approximation (nearly indistinguishable for $\eta>5$). So $F(\eta)$ oscillates around 1, with an amplitude that decays as $\eta^{-3/4}$. The maximum 1.5487 is reached at $\eta= 3.37213$.

Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2/\pi}\frac{1}{\eta^{3/4}} \left[\sin \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)\right],\;\;\eta\gg 1.$$

see plot, blue is $F(\eta)$, gold is the large-$\eta$ approximation (nearly indistinguishable for $\eta>5$). So $F(\eta)$ oscillates around 1, with an amplitude that decays as $\eta^{-3/4}$. The maximum 1.5487 is reached at $\eta= 3.37213$.

This representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2\pi}\frac{1}{\eta^{3/4}} \left[\sin \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)\right],\;\;\eta\gg 1.$$

see plot, blue is $F(\eta)$, gold is the large-$\eta$ approximation (nearly indistinguishable for $\eta>5$). So $F(\eta)$ oscillates around 1, with an amplitude that decays as $\eta^{-3/4}$. The maximum 1.5487 is reached at $\eta= 3.37213$.

Nemo's representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2\pi}\frac{1}{\eta^{3/4}} \left[\sin \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)\right],\;\;\eta\gg 1.$$

see plot, blue is $F(\eta)$, gold is the large-$\eta$ approximation (nearly indistinguishable for $\eta>5$). So $F(\eta)$ oscillates around 1, with an amplitude that decays as $\eta^{-3/4}$. The maximum 1.5487 is reached at $\eta= 3.37213$.

$$F(\eta)=-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}+\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2\pi}\frac{1}{\eta^{3/4}} \left[\sin \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)\right],\;\;\eta\gg 1.$$

see plot, blue is $F(\eta)$, gold is the large-$\eta$ approximation (nearly indistinguishable for $\eta>5$). So $F(\eta)$ is not strictly above 1, it oscillates around 1, with an amplitude that decays as $\eta^{-3/4}$. The maximum 1.5487 is reached at $\eta= 3.37213$.

This representation of $F(\eta)$ in terms of a hypergeometric function can be evaluated without difficulty for large $\eta$: $$F(\eta)=\frac{\sqrt{3} {\eta}^2 \Gamma \left(\frac{2}{3}\right) \; _1F_2\left(\frac{2}{3};\frac{4}{3},\frac{5}{3};-\frac{{\eta}^3}{27}\right)}{6\pi }-\frac{12 {\eta} \; _1F_2\left(\frac{1}{3};\frac{2}{3},\frac{4}{3};-\frac{{\eta}^3}{27}\right)}{6\Gamma \left(-\frac{1}{3}\right)}-\frac{1}{3}$$

$$F(\eta)\rightarrow 1+\sqrt[4]{3} \sqrt{2\pi}\frac{1}{\eta^{3/4}} \left[\sin \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)-\cos \left(\frac{2 \eta^{3/2}}{3 \sqrt{3}}\right)\right],\;\;\eta\gg 1.$$

see plot, blue is $F(\eta)$, gold is the large-$\eta$ approximation (nearly indistinguishable for $\eta>5$). So $F(\eta)$ oscillates around 1, with an amplitude that decays as $\eta^{-3/4}$. The maximum 1.5487 is reached at $\eta= 3.37213$.