$\newcommand\P{\mathcal P}$A "partition" $P$ (of the interval $[0,1]$) is a finite sequence $(t_0,\dots,t_n)$ such that $0=t_0<\cdots<t_n=1$; then the mesh of $P$ is $\|P\|:=\max_{1\le j\le n}(t_j-t_{j-1})$.

Fix any sequence $\P:=(P_k)$ of "partitions" $P_k=(t_{k,0},\dots,t_{k,n_k})$ such that $\|P_k\|\to0$ (as $k\to\infty$).

For a real-valued function $f$ on $[0,1]$, define its quadratic variation (with respect to $\P$) by the formula $$[f]_\P:=\limsup_{k\to\infty}\sum_{j=1}^{n_k}(f(t_{k,j})-f(t_{k,j-1}))^2.$$

Suppose now that $f$ is differentiable and $[f]_\P<\infty$. Does it then necessarily follow that $[f]_\P=0$?

This question is a modification of the (now answered, positively) previous question A dichotomy for the quadratic variation of differentiable functions?. The difference is that now the sequence $\mathcal P$ is fixed.