Linearity of motivic Hirzebruch class transformations

Let $X$ be a complex algebraic variety. Consider a natural transformation

$$T:K_0(\operatorname{VAR}/X)\to H_*^{BM}(X)\otimes\mathbb Q[y]$$

without the normalization condition on smooth varieties. Let ${{td_y}_*}_i$ be the degree-$2i$ component of the motivic Hirzebruch class transformation. Linearity conjecture. Every such natural transformation is a linear combination

$$T=\sum_{i\geq 0}r_i(y)\,{{td_y}_*}_i, \qquad r_i(y)\in\mathbb Q[y].$$

This conjecture would mean that the normalization condition used to construct the motivic characteristic class can essentially be dropped. It is motivated by corresponding classification theorems for Todd and MacPherson Chern class transformations, while the asserted classification in the motivic Hirzebruch setting remains open.