Linearity of Cappell-Shaneson L-class transformations

Let $\Omega(-)$ denote the domain of Cappell--Shaneson's L-class transformations, and consider a natural transformation

$$T:\Omega(-)\to H_*^{BM}(-)\otimes\mathbb Q$$

without the normalization condition. Let ${L_*^{CS}}_i$ denote the degree-$2i$ component of the Cappell--Shaneson L-class transformation. Cappell-Shaneson linearity conjecture. Every such natural transformation is a linear combination

$$T=\sum_{i\geq 0}r_i\,{L_*^{CS}}_i, \qquad r_i\in\mathbb Q.$$

The conjecture is proposed as the analogue of known linearity results for Todd and rationalized MacPherson Chern class transformations; the source provides no resolution of this L-class classification problem.