The Pollyanna conjecture for box intersection graphs

Fix a positive integer $d$, and let $\mathcal{B}_d$ be the class of intersection graphs of axis-aligned boxes in $\mathbb{R}^d$. A hereditary class is Pollyanna if every hereditary χ-bounded subclass of it is polynomially χ-bounded. Pollyanna conjecture for box intersection graphs. The class $\mathcal{B}_d$ is Pollyanna. This would generalize the known polynomial χ-boundedness results for interval graphs and rectangle intersection graphs; the source presents it as a possible broader principle and does not state a resolution.