Liverpool meet West Ham in London on Sunday on the back of nine defeats in their past 12 games.
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"We have to take this moment on the chest. Criticism is part of football, and we'll fight back every time and come through this storm," Konate wrote on Instagram.
"In these moments, the fans matter most. The ones who stand with us in the good times and the bad, the ones who sing for us even in the difficult moments.
"Your voice and your support mean everything to us. We know we need to improve and we'll keep fighting for you. Always."