I asked this question on Mathematics Stack Exchange but got no answer.

Here is the question:

Let $A$ be a domain (that is, a commutative ring with one in which the condition $ab=0$ implies $a=0$ or $b=0$).

Assume that $A$ has the following property:

If $\mathfrak p_1,\dots,\mathfrak p_k$ are distinct nonzero prime ideals of $A$, and if $m$ and $n$ are distinct elements of $\mathbb N^k$, then we have $$ \mathfrak p_1^{m_1}\cdots\mathfrak p_k^{m_k}\ne\mathfrak p_1^{n_1}\cdots\mathfrak p_k^{n_k}. $$

Does it follow that $A$ is locally noetherian?

Here are a couple of comments:

In this answer Badam Baplan pointed out that locally noetherian domains do have the indicated property, and that that some *non-noetherian* domains are *locally* noetherian (the first example seems to have been given by N. Nakano in 1953).

Previously user26857 had observed that noetherian domains have this property, and Julian Rosen had shown that many non-noetherian domains do *not* have the property.