The transition-map conjecture for motivic Białynicki–Birula decompositions

Let $C$ be a curve and let $D'\geq D$ be effective divisors on $C$. For the motivic Białynicki–Birula decompositions, write $C^{(\underline m)}$ for the relevant product of symmetric powers, $c_{\underline m}$ for its associated codimension shift, and let

$$f_{\underline m,\underline m'}:C^{(\underline m)}\longrightarrow C^{(\underline m')}$$

be the morphism that adds $n(D'-D)$ to the first symmetric-power factor and is the identity on the remaining factors. The induced morphisms $k_{\underline m,\underline m'}$ fit into the stated commutative diagram. The transition-map conjecture. The morphisms

k_{\underline m,\underline m'}:M(C^{(\underline m)})\{c_{\underline m}\}\longrightarrow M(C^{(\underline m')})\{c_{\underline m'}\}\

are $M(f_{\underline m,\underline m'})\{c_{\underline m}\}$ when $\underline m'=\underline m+(n\deg(D'-D),0,\ldots,0)$, and are zero otherwise. This conjecture describes the functoriality of the motivic Białynicki–Birula decompositions under inclusions of divisor bounds; the supplied source gives no resolution.