Consider the branched covering $f \colon X \to \mathcal{Q}_7$ of the $7$-dimensional smooth projective quadric by a smooth connected projective variety $X$. Since we have the $6$-dimensional quadric $\mathcal{Q}_6$ as codimension $1$ subvariety in $\mathcal{Q}_7$ as a smooth hyperplane section, we get a divisor $H \subset X$, which is the pre-image of $\mathcal{Q}_6$ in $X$.
Consider the canonical bundle $K_X$. We get a vector bundle $f_*K_X$ on $\mathcal{Q}_7$, which can be restricted to $\mathcal{Q}_6$. On the other hand, we could restrict $K_X$ to the divisor $H$ and apply pushforward $f_*(K_X|_H).$
My question is how to check, whether $$ f_*(K_X)|_{\mathcal{Q}_6} = f_*(K_X|_H)$$ or if there's even a chance that this equality exits. Thanks in advance.