Bell-polynomial formula for multiple-point corrections

Let $S$ be the surface, $\mathscr{L}$ the line bundle, and $Y=|\mathscr{L}|$ of dimension $N$. Let $f$ be the relevant map, let $M_i(S,\mathscr{L})$ be the classes defined by

$$M_r(S,\mathscr{L})=c(N_XF)^{r-1}(\iota^*\nu^*c(T_S))^{r-1}\cap[X],$$

let $Q_i$ be obtained from the component of $M_i(S,\mathscr{L})$ of dimension $N-i$, and let $P_r$ denote the $r$th complete Bell polynomial. Bell-polynomial correction conjecture. For $r\geq1$, there is a $\mathbb{Q}$-linear combination $C_r$ of the integers

$$\int_Y f_*\\{M_i(S,\mathscr{L})\\}_{N-r},$$

for $2\leq i\leq r-1$, with $C_1=C_2=0$, such that

$$\int_Y f_*m_r=P_r(Q_1+C_1,-2(Q_2+C_2),\ldots,(-1)^{r-1}(r-1)!(Q_r+C_r)).$$

The claim seeks a systematic Bell-polynomial expression for the multiple-point contributions, with correction terms accounting for Segre-class inclusion--exclusion. It is presented only as evidence-supported and no resolution is given in the supplied text.