Centered maximal variation conjecture with endpoint improvement

Let $M$ be the centered Hardy--Littlewood maximal operator on $fR$, and let $f\in\operatorname{BV}(fR)$ have limits $f(\infty)$ and $f(-\infty)$ at the two ends. Write $\operatorname{Var}(f)$ for the total variation of $f$. Centered maximal variation conjecture with endpoint improvement. One should have

$$\operatorname{Var}(Mf)\leq \operatorname{Var}(f)-\frac12\big|\,|f(\infty)|-|f(-\infty)|\,\big|.$$

This would strengthen the conjectured bound $\operatorname{Var}(Mf)\leq\operatorname{Var}(f)$ for the centered operator, improving the known estimate with an unspecified constant factor. The paper proves the displayed inequality for piecewise constant functions with alternating nonzero and zero values.