Stabilisation of the LHS spectral sequence for algebraic groups

In this note, we consider the Lyndon-Hochschild-Serre spectral sequence corresponding to the first Frobenius kernel of an algebraic group G and computing the extensions between simple G-modules. We state and discuss a conjecture that E2 = E and provide general conditions for low-dimensional terms on the E2-page to be the same as the corresponding terms on the E∞-page, i.e. its abutment.