Polynomial-time clustering threshold for Gaussian mixtures

Let $n$, $d$, and $K$ be positive integers with $d\geq K$, and let $\Delta_*$ denote the separation parameter for the Gaussian-mixture clustering model. Write $a\gtrsim_{\log} b$ when $a$ is at least $b$ up to polylogarithmic factors. Polynomial-time clustering conjecture. For any $n$, $d$, and $K$ such that $d\geq K$, it is possible to reconstruct the partition in polynomial time as long as

$$\Delta_*^2 \gtrsim_{\log} \left[1 + \left( \sqrt{\frac{dK^2}{n}}+ \frac{K}{n^{1/4}} \right) \wedge \sqrt{d} \right].$$

Together with the paper's low-degree lower bound and the lower bounds of Even (2025), this would identify the computational threshold up to polylogarithmic factors for the stated range of dimensions. The paper proves a matching upper bound only for specific instances, so the general assertion remains open.