Derived Satake morphisms for $p$-small weights in characteristic $p$

Let $F$ be a finite unramified extension of $\mathbb{Q}_p$ with ring of integers $\mathcal{O}_F$, and let $\mathbf{G}$ denote a split, connected reductive group over $\mathcal{O}_F$. We fix a Borel subgroup $\mathbf{B} = \mathbf{T}\mathbf{U}$ with maximal torus $\mathbf{T}$ and unipotent radical $\mathbf{U}$, and let $L(\lambda)$ denote an irreducible representation of $G_0 := \mathbf{G}(\mathcal{O}_F)$ with coefficients in a sufficiently large field of characteristic $p$. Set $G := \mathbf{G}(F)$, etc. Assuming $\lambda$ is a $p$-small and sufficiently regular character and that $p - 1$ is greater than the Coxeter number of $\mathbf{G}$, we show that the complex $L(U,\textrm{c-ind}_{G_0}^{G}(L(\lambda)))$ splits as the orthogonal direct sum of its cohomology objects in the derived category of smooth $T$-representations in characteristic $p$. (Here $L(U, -)$ denotes Heyer's left adjoint of parabolic induction, from the derived category of smooth $G$-representations to the derived category of smooth $T$-representations.) Consequently, this gives rise to a collection of morphisms of graded spherical Hecke algebras $$\displaystyle{\bigoplus_{i \in \mathbb{Z}}\textrm{Ext}_{G}^{i}\left(\textrm{c-ind}_{G_0}^{G}(L(\lambda)),~\textrm{c-ind}_{G_0}^{G}(L(\lambda))\right) \longrightarrow \bigoplus_{i \in \mathbb{Z}}\textrm{Ext}_{T}^{i}\left(\textrm{c-ind}_{T_0}^{T}(L^n(U_0,L(\lambda))),~\textrm{c-ind}_{T_0}^{T}(L^n(U_0,L(\lambda)))\right)}$$ indexed by $n=-[F:\mathbb{Q}_p]\dim(\mathbf{U}), \ldots, 0$, which we refer to as derived Satake morphisms. For $\lambda=0$ and $n=0$, this recovers the graded mod $p$ Satake homomorphism constructed by Ronchetti. We also give some partial results for general standard parabolic subgroups $\mathbf{P} = \mathbf{M}\mathbf{N} \subset \mathbf{G}$.