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Let $R$ be a non-noetherian local domain. Suppose that the following two conditions hold for $R$$\colon$

$(*)$$~\quad$An arbitrary prime ideal ${\frak P}$ of $R$ such that ${\mathrm{ht}}({\frak P}) < \infty$ is finitely-generated.

$(**)$$\!\quad$An arbitrary finitely-generated ideal $I$ of $R$ satisfies that ${\mathrm{ht}}(I) < \infty$.

Q. Is $R$ necessarily coherent?

That is, the intersection $I \cap J$ of any two finitely-generated ideal $I$ and $J$ of $R$ is also finitely-generated.

The $R$ in study above seems quite similar to noetherain rings. I cannot, however, find any references on the question.

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