Deformation to the associated graded algebra via an injective well-ordered valuation

Let $A$ be a domain equipped with an injective well-ordered valuation $\nu$, and let $gr A$ denote the associated graded algebra of the filtration on $A$ induced by $\nu$. A family of algebras $A_t$ is said to connect $A$ to $gr A$ if it is defined for the relevant parameter values and satisfies the stated specializations. Valuation deformation conjecture. For every such pair $(A,\nu)$, there is a family $A_t$ such that

$$A_1=A,\qquad A_0=gr A.$$

The preceding proposition establishes this construction for one-dimensional finitely generated algebras presented as $\Bbbk[x_1,\dots,x_n]/J$ under its stated hypotheses, suggesting the broader formulation for arbitrary domains with injective well-ordered valuations.