This is a follow up of this question.

Let $U_q\mathfrak{sl}(2)$ be Lusztig's integral form of the quantized enveloping algebra of $\mathfrak{sl}_2$, specialised at $q$ a root of unity. This is an algebra generated by elements $E,F,K$, and by the divided powers of $E$ and $F$. Let $\mathcal O$ be its category of finite dimensional (type $I$, integrable) representations.

Is it true that every indecomposable object $M\in\mathcal O$ has weight spaces that are at most $2$-dimensional?

(I know that this is true for tilting modules.)


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