Fredholm range and kernel conjecture for the coagulation-fragmentation operator

Let $\operatorname{R}(W_{\phi})$ and $\operatorname{N}(W_{\phi})$ denote the range and null space of the operator $W_{\phi}$ on $L^1([0,L])$, and let $\operatorname{span}\{x\}^{\perp}$ be the orthogonal complement of the span of the function $x$. Fredholm range and kernel conjecture.

$$\operatorname{R}(W_{\phi})=\operatorname{span}\{x\}^{\perp},\qquad \dim\operatorname{N}(W_{\phi})=1,$$

and

$$\operatorname{N}(W_{\phi})\cap\operatorname{span}\{x\}^{\perp}=\{0\}.$$

The conjecture is motivated by Fredholm theory and strengthens the preceding inclusion of the range in $\operatorname{span}\{x\}^{\perp}$; no resolution is given in the source.