= Solution
The given large-angle temperature transfer relation gives the <low-multipole tensor fraction in quartic hilltop inflation>
$$
\boxed{\frac TS\simeq-7n_T
\simeq3.68\times10^{-4}\left(\frac{\sigma}{m_{\rm pl}}\right)^4\ll10^{-4}.}
$$
This is not the primordial <tensor-to-scalar ratio> $r=16\epsilon_V$, because the scalar and tensor contributions to low CMB multipoles have different transfer functions. With $\sigma\ll m_{\rm pl}$, the tensor contribution is far below the scalar temperature signal and its <cosmic variance>, and the corresponding primordial gravitational-wave or polarization signal is extremely small.
For <COBE normalization of quartic hilltop inflation>, the historical scalar horizon amplitude is of order $\delta_H\simeq2\times10^{-5}$, or $\mathcal P_S\simeq25\delta_H^2/4\sim2.5\times10^{-9}$ in the usual matter-era convention. The slow-roll spectrum simplifies to
$$
\mathcal P_S\simeq\frac{V^3}{12\pi^2M^6V'^2}
\simeq\frac{\alpha\sigma^{12}}{1728\pi^2M^6\phi_{50}^6}
=\frac{8\alpha(50)^3}{\pi^2}.
$$
Hence
$$
\boxed{\alpha\simeq\frac{\pi^2\mathcal P_S}{8(50)^3}\sim2\text{--}3\times10^{-14}.}
$$
Late-time transfer and tilt corrections to the COBE normalization change the order-one coefficient, not the conclusion that the dimensionless coupling must be extraordinarily small. The leading normalization is independent of $\sigma$: reducing the height of the potential also changes the slope and field value at fixed $N$, causing the scale factors to cancel.
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