= Solution
Fix the specified time $t$. With the path modulus $\eta_n$ on $[0,t+1]$ as above, comparison of increment powers now gives
$$
V_t^{n,|\cdot|^2}\le\eta_n^{2-p}V_t^{n,|\cdot|^p}.
$$
The hypothesis makes the second factor eventually bounded <almost surely>, while the first tends to zero <almost surely> because $2-p>0$. Thus the squared-increment sums tend to zero <almost surely>. They also converge in <probability> to the <quadratic variation> $[X]_t$, so uniqueness of the limit in <probability> implies $[X]_t=0$ <almost surely>.
The <quadratic variation> is nondecreasing, hence it vanishes throughout $[0,t]$. To see explicitly that the <continuous local martingale> must then be constant, localize it to square-integrable stopped <martingales>. The <Itô isometry> gives
$$
\mathbb E\bigl|X_{s\wedge\sigma_N}-X_0\bigr|^2
=\mathbb E[X]_{s\wedge\sigma_N}=0\qquad(s\le t).
$$
Remove the localization, apply the conclusion at every rational $s\le t$, and use continuity. Since $X_0=0$, this proves
$$
\boxed{\mathbb P(X_s=0\text{ for every }s\in[0,t])=1.}
$$
Thus $X$ is indistinguishable from the zero process on that interval. This is the subquadratic case of <dyadic power variation of a continuous local martingale>.
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