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, computing the extensions between simple $G$-modules. We state and discuss a conjecture that $E_2=E_{\mathrm{\infty }}$ and provide general conditions for low-dimensional terms on the $E_2$-page to be the same as the corresponding terms on the $E_{\mathrm{\infty }}$-page, i.e. its abutment.