The sharp maximal-variation inequality on the integer lattice

Let $p e(1/2,1]$ and let $f:\mathbb{Z}\to\mathbb{R}$ be a function in $ell^{p}(\mathbb{Z})$. Write ${\rm Var}_p f$ for the $p$-variation of $f$, and let $Mf$ denote the centered discrete Hardy–Littlewood maximal function on $\mathbb{Z}$. Sharp maximal-variation conjecture. One has

$${\rm Var}_p Mf\leq \left(\sum_{k=0}^{\infty}\frac{2^{p}}{(2k+1)^p(2k+3)^p}\right)^{\frac{1}{p}}{\rm Var}_pf.$$

The preceding discussion establishes the analogous sharp bound for the maximal operator acting on functions when the variation exponent is in the complementary range described by the paper, while the displayed assertion is posed as the remaining question for $p\in(1/2,1]$.