Characterization of local rings whose only strong Rees ideals are powers of the maximal ideal

Let $(A,\mathfrak{m})$ be a local ring of dimension $d\geq 2$. An ideal has the strong Rees property if it satisfies the property studied in the paper, and suppose that the only ideals of $A$ with this property are the powers $\mathfrak{m}^n$.

Strong Rees property characterization conjecture. Then $\dim A=2$, and either $A$ is a regular local ring or

$$\widehat{A}\cong k[[X^r,X^{r-1}Y,\ldots,Y^r]]$$

for some $r$.

This conjecture asks whether the strong Rees property characterizes regular local rings and the indicated Veronese subrings in dimension two. The supplied text gives no resolution, so the conjecture remains open.