Kuznetsov's Fano threefold conjecture via Hochschild-Serre algebra

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.