The path-graph variational constant conjecture

Let $P_n$ be the path graph on $n$ vertices, and let $\mathbf{C}_{P_n}$ denote its optimal $1$-variational constant, namely the smallest constant such that

$$\textup{Var} M_{P_n}f\leqslant \mathbf{C}_{P_n}\textup{Var} f$$

for every real-valued function $f$ on $P_n$.

Path-graph conjecture. For every integer $n\geqslant3$,

$$\mathbf{C}_{P_n}=1-\frac1n.$$

The source reports numerical evidence for all $n\leqslant10$. The conjecture would imply the bound $\mathbf{C}_{\mathbb{Z}}\leqslant1$ for the integer lattice, and hence the corresponding real-line bound, but it remains open and is expected to be difficult.