Equidimensional equality criterion for Serre intersection multiplicity

Let $A$ be a regular local ring. Let $M$ and $N$ be equidimensional, finitely generated $A$-modules such that

$$\dim M+\dim N=\dim A,$$

and $\ell(M\otimes_A N)<\infty$. Write $\operatorname{gr}A$ for the associated graded ring and $\operatorname{gr}M$, $\operatorname{gr}N$ for the associated graded modules. Equidimensional equality criterion. Then

$$\chi^A(M,N)\geq e(M)e(N),$$

and equality occurs if and only if

$$\dim\left(\operatorname{gr}M\otimes_{\operatorname{gr}A}\operatorname{gr}N\right)=0.$$

This is presented as the algebraic form of the stronger positivity conjecture for strict transforms on the blowup. The supplied text does not state that it is known or resolved.