Nori's Homotopy Conjecture for lifting surjections

Let $X=\operatorname{Spec}(A)$ be a smooth affine variety of dimension $d$. Let $P$ be a projective $A$-module of rank $r$, and let $f_0:P\twoheadrightarrow I$ be a surjective homomorphism onto an ideal $I$ of $A$. Assume that $Y=V(I)$ is smooth of dimension $d-r$. Let $Z=V(J)\subseteq \operatorname{Spec}(A[T])=X\times \mathbb{A}^1$ be a smooth subscheme intersecting $X\times\{0\}$ transversally in $Y\times\{0\}$. Suppose that $\varphi:P[T]\twoheadrightarrow J/J^2$ is a surjective map satisfying

$$\varphi_{|T=0}=f_0\otimes \frac{A}{I}.$$

Homotopy Conjecture. There exists a surjective map $F:P[T]\twoheadrightarrow J$ such that

$$F_{|T=0}=f_0$$

and

$$F_{|Z}=\varphi.$$

The conjecture is a lifting statement for homotopy obstructions associated with surjections onto ideals; it asserts that compatible data modulo $J^2$ can be lifted while preserving the prescribed specialization at $T=0$. The source presents it as Nori's Homotopy Conjecture, while noting that Nori's ideas were communicated informally and appeared in differing forms in the literature; its resolution is not established in the supplied text.