The chromatic quasisymmetric function conjecture for directed trees

Let $\overrightarrow{G}$ and $\overrightarrow{H}$ be directed trees, and let $X_{\overrightarrow{G}}(\mathbf{x},t)$ denote the chromatic quasisymmetric function of a directed graph. The chromatic quasisymmetric function conjecture. If $\overrightarrow{G}$ and $\overrightarrow{H}$ are not isomorphic, then

$$X_{\overrightarrow{G}}(\mathbf{x},t)\neq X_{\overrightarrow{H}}(\mathbf{x},t).$$

This is a natural quasisymmetric analogue of Stanley's conjecture that the chromatic symmetric function distinguishes trees; it was previously stated as a question by Alexandersson and Sulzgruber. The conjecture is open.