Conjectural unstable Poincaré–Hopf formula at non-rational points

Let $f/g:\mathbb{P}^1\to\mathbb{P}^1$ be a pointed rational function with vanishing locus $D=\{r_1,\ldots,r_n\}\subset\mathbb{A}^1_k$. Let $m_i(x)\in k[x]$ denote the monic minimal polynomial of $r_i$ for each $i$. At each zero, let the unstable local degree be

$$\deg_{r_i}(f/g)=(\beta_i,d_i)\in\operatorname{GW}(k)\times_{k^\times/(k^\times)^2}k^\times,$$

where $\operatorname{GW}(k)$ denotes the Grothendieck–Witt group over $k$. Unstable Poincaré–Hopf conjecture. The unstable degree of $f/g$ should decompose as

$$\deg(f/g)=\left(\bigoplus_{i=1}^n\beta_i,\prod_{i=1}^n d_i\cdot\operatorname{tol}\left(\prod_{i=1}^n m_i(x)\right)\right).$$

This conjecture removes the rationality assumption from the corresponding Poincaré–Hopf formula and predicts the correction contributed by the minimal polynomials of the non-rational zeros through the tolerant of their product. Its status is not established in the supplied text.