Matkowski's piecewise-linear conjecture for continuous quasi graph-additive functions

Let $f:\mathbb{R}\to\mathbb{R}$ be continuous. The equation under consideration is

$$f\big(f(-x)+x\big)=f\big(-f(x)\big)+f(x),\qquad x\in\mathbb{R}.$$

Matkowski's conjecture. The function $f$ solves this equation if and only if there exist $a,b\in\mathbb{R}$ with $a\neq b$ such that

$$f(x)= \begin{cases} ax&\text{if }x\leq 0,\\ bx&\text{if }x>0, \end{cases}$$

where $a,b\in[0,1]$ or $a+b=1$. This conjecture concerns the classification of continuous solutions and is presented as a motivation for the counterexample in the paper; its resolution is not specified in the supplied text.