Tensor and lavish support conjecture for arbitrary almost-simple groups

Let $G$ be an arbitrary almost-simple algebraic group, and let

$$\lambda:\operatorname{Spec}(K)\to \widecheck{G}/\widecheck{B}$$

be an arbitrary geometric point. Let $\mathscr{B}_\lambda$ be the corresponding object, and let $\operatorname{stab}(\mathscr{B}_\lambda)$ denote its stable endomorphism algebra. Cohomological support is the support theory associated to this algebra.

Standard conjecture. The following assertions hold:

1. Cohomological support for $\operatorname{stab}(\mathscr{B}_\lambda)$ satisfies the tensor product property.
2. Cohomological support for $\operatorname{stab}(\mathscr{B}_\lambda)$ is a lavish support theory.

These properties would extend the support-theoretic results to arbitrary almost-simple Dynkin type. The source says that this statement was previously conjectured, but provides no resolution evidence here.