Strong openness of multiplier ideal sheaves and optimal $L^2$ extension

In this note, we reveal that our solution of Demailly’s strong openness conjecture implies a matrix version of the conjecture; our solutions of two conjectures of Demailly-Koll'{a}r and Jonsson-Mustat\u{a} implies the truth of twisted versions of the strong openness conjecture; our optimal $L^2$ extension implies Berndtsson’s positivity of vector bundles associated to holomorphic fibrations over a unit disc.