Monodromy of Tate-Linear varieties

We answer a question of Chai about the rigidity of Tate-linear subvarieties of ${\mathcal{A}}_g$ over ${\bar{\mathbb{F}}}_p$: we show that that a smooth subvariety of a central leaf of ${\mathcal{A}}_g$ which is Tate-linear at one point is Tate-linear at every point. This is achieved by a careful analysis of the $p$-adic monodromy groups of Tate-linear varieties, building on recent work of D’Addezio and van Hoften.