= Solution
The initial field values are $\phi_i/M_{\rm Pl}=\sqrt{122n}$, approximately $15.6$, $19.1$ and $22.1$ for the three powers. A simple <monomial inflation potential> over this super-Planckian field range needs theoretical protection. Generic <Planck-suppressed corrections to inflaton potentials>, such as $c_p\phi^p/M_{\rm Pl}^{p-4}$, need not be small there; they can change the slope and curvature and spoil <slow-roll inflation>. <Radiative corrections> and possible couplings to other particles likewise require control. A symmetry, such as an approximate <scalar-field shift symmetry>, or a specified <ultraviolet completion> could supply that protection, but it is not part of the bare monomial model.
A large field value is not by itself a proof that the <energy density> is Planckian: a sufficiently small $\lambda$ can keep $V\ll M_{\rm Pl}^4$. The issue is control of the <effective field theory> and stability of the flat potential over its field range, rather than simply comparing the field value with a mass scale.
There is also a global potential issue for $n=3$: $V=\lambda\phi^3$ continued over all real $\phi$ is unbounded below and has no stable minimum at zero. Restricting to $\phi>0$ does not specify what happens when the field reaches that boundary, so a completion is needed for post-inflationary evolution and <reheating>. The even powers have a stable minimum but still need interactions that transfer the inflaton energy to a hot bath. These interactions and the resulting <reheating> history also affect the mapping between a pivot scale and the assumed 60 <number of e-folds>. Therefore the concise theoretical concerns are \b[control of large-field corrections, a consistent stable completion, and a specified reheating mechanism].
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