Loewy-length conjecture for projective modules in standard Levi form

Let $\chi$ be a $p$-character in standard Levi form with $I=\Pi$, let $w_I$ be the corresponding longest Weyl-group element, and let $R_I^+$ be the associated set of positive roots. For a $p$-regular weight $\lambda$, let $\widehat Q_{\chi,I}(\lambda)$ be the projective cover of the irreducible baby Verma module and let $\mathcal L_\chi(\lambda)$ be the associated quasi-simple module.

Projective-module Loewy-length conjecture. If $\chi$ has standard Levi form with $I=\Pi$ and $\lambda$ is $p$-regular, then

$$ll(\widehat Q_{\chi,I}(\lambda))=ll(\mathcal L_\chi(\lambda))=l(w_I)+1=|R_I^+|+1.$$

This gives a precise uniform prediction for the Loewy lengths of projective modules and quasi-simple modules in the standard-Levi case, extending the motivation from known special cases.