  Let $Y$ be a smooth quartic double solid regarded as a degree 4 hypersurface
of the weighted projective space $\mathbb{P}(1,1,1,1,2)$. We study the
multiplication of Hochschild-Serre algebra of its Kuznetsov component
$\mathcal{K}u(Y)$, via matrix factorization. As an application, we give a new
disproof of Kuznetsov's Fano threefold conjecture. In appendix, we show kernel
of differential of period map for special Gushel-Mukai threefold is of two
dimensional by categorical methods, which completes a result in \cite[Theorem
7.8]{debarre2008period}.
