The norm in $C$ of orthogonal projections onto subspaces of polygonal functions

Let $P_\pi$ be an orthogonal projection (in the sense of $L_2$) onto the subspace of polygonal functions over a certain partition $\pi$ of the segment $[0,1]$. Z. Ciesielski has established the following estimate for the norm of this operators, as acting from $C$ into $C$, valid for an arbitrary partition: $\|P_\pi\|_{C\to C}\le3$. In this note it is proved that this estimate is final; more precisely, it is shown that $\sup\limits_\pi\|P_\pi\|_{C\to C}=3$.