The core formula for ideals generated by general forms

Let $R=k[x_1,\ldots,x_n]_{(x_1,\ldots,x_n)}$ with $k$ a field of characteristic zero, and let $\mathfrak{m}$ denote the maximal ideal of $R$. Let $I=I_d+\mathfrak{m}^{d+1}$, where $I_d$ is a complete intersection ideal of $s$ general $d$-forms with $1\leq s<n$. Define

$$b=\left\lfloor\frac{dn-s+1}{d+1}\right\rfloor,$$

and

$$a=dn-s+1-(d+1)b.$$

Core formula conjecture. The core of $I$ is

$$\operatorname{core}(I)=\mathfrak{m}^{a}I^{b}.$$

The conjecture proposes a general formula for the core of this family of integrally closed ideals, extending the motivating example and addressing the failure of core to preserve inclusion. Its resolution is not specified in the supplied text.