Conjecture on Loewy lengths of standard and projective modules

Let $\chi$ be a $p$-character in standard Levi form, let $I$ be the associated subset of simple roots, and let $w_I$ and $w^I$ denote the corresponding Weyl-group elements. For $\lambda\in X(T)$ that is $p$-regular, let $\widehat Z_\chi(\lambda)$ and $\widehat Z^{w^I}_\chi(\lambda^{w^I})$ be the proper standard modules, let $\widehat Q^I_\chi(\lambda)$ and $\widehat Q^{w^I}_\chi(\lambda^{w^I})$ be the standard modules, and let $\widehat Q_\chi(\lambda)$ be the indecomposable projective module.

Loewy-length conjecture. Assume that $\chi$ has standard Levi form. When $\lambda\in X(T)$ is $p$-regular,

$$ll(\widehat Z_\chi(\lambda))=ll(\widehat Z^{w^I}_\chi(\lambda^{w^I}))=l(w^I)+1,$$ $$ll(\widehat Q^I_\chi(\lambda))=ll(\widehat Q^{w^I}_\chi(\lambda^{w^I}))=l(w^I)+l(w_I)+1,$$

and

$$ll(\widehat Q_\chi(\lambda))=2l(w^I)+l(w_I)+1.$$

These formulas summarize the proposed Loewy lengths for proper standard, standard, and indecomposable projective modules. They are motivated by the preceding results, while their general validity remains open.