The induced-grid conjecture for tree-independence number

Let $d$ be a positive integer and let $\mathcal{C}$ be a hereditary graph class excluding $K_{1,d}$. Define its induced minor closure by

$$\widehat{\mathcal{C}}=\{H\mid H\text{ is an induced minor of some }G\in\mathcal{C}\}.$$

The induced-grid conjecture. The class $\mathcal{C}$ has bounded tree-independence number if and only if $\widehat{\mathcal{C}}$ does not contain all planar graphs. This is proposed as a generalization of the induced Grid-Minor Theorem from bounded-degree graphs to hereditary classes excluding a star; the source presents it as open.