Spin character-square conjecture for double covers of alternating groups

Let $\widetilde{A}_n$ be the double cover of the alternating group $A_n$, and let $\operatorname{Irr}(\widetilde{A}_n)$ denote its irreducible characters. A character is called spin when it is faithful, and a character is non-faithful when it factors through $A_n$. Spin character-square conjecture. For any $n\geq 5$, there is a spin character $\chi\in\operatorname{Irr}(\widetilde{A}_n)$ whose square $\chi^2$ contains all non-faithful $\psi\in\operatorname{Irr}(\widetilde{A}_n)$. GAP data through $n=25$ provides evidence, and the source notes that the proportion of spin characters with the stated property appears to grow with $n$; the general assertion remains open.