= Solution
Fix $T<\infty$. By <uniform continuity> of each path of the <continuous local martingale> on $[0,T+1]$, its modulus
$$
\eta_n=\sup_{\substack{u,v\in[0,T+1]\\|u-v|\le h_n}}|X_u-X_v|
$$
tends to zero <almost surely>. For every $s\le T$, comparison of powers of each increment gives
$$
V_s^{n,|\cdot|^p}\le\eta_n^{p-2}V_T^{n,|\cdot|^2}.
$$
The squared-increment sum converges in <probability> to $[X]_T$, by the dyadic definition of <quadratic variation>. Including the last full grid interval instead of the last partial interval changes that sum by a quantity bounded by $2\eta_n^2$, so the ceiling convention has the same limit. In particular these squared-increment sums are bounded in <probability>.
Since $p-2>0$, a quantity tending to zero <almost surely> multiplied by a family bounded in <probability> tends to zero in <probability>. Hence
$$
\boxed{\sup_{s\le T}V_s^{n,|\cdot|^p}\xrightarrow{\mathbb P}0\quad\text{for every }T<\infty.}
$$
This is <uniform convergence on compacts in probability>, and proves the superquadratic case of <dyadic power variation of a continuous local martingale>.
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