Tensoring generalized characters with the Steinberg character

Let $\mathbf{G}$ be a reductive connected algebraic group over an algebraic closure of a finite field of characteristic $p$. Let $F$ be a Frobenius endomorphism on $\mathbf{G}$ and write $G := \mathbf{G}^F$ for the corresponding finite group of Lie type.

We consider projective characters of $G$ in characteristic $p$ of the form $St \cdot \beta$, where $\beta$ is an irreducible Brauer character and $St$ the Steinberg character of $G$.

Let $M$ be a rational $\mathbf{G}$-module affording $\beta$ on restriction to $G$. We say that $M$ is $G$-regular if for every $F$-stable maximal torus $\mathbf{T}$ distinct weight spaces of $M$ are non-isomorphic $\mathbf{T}^F$-modules. We show that if $M$ is $G$-regular of dimension $d$, then the lift of $St \cdot \beta$ decomposes as a sum of $d$ regular characters of $G$.