Orthogonality in QR Decomposition

Question

Given $L\in\mathbb{R}^{m\times n}$, I define $Y_2=(L^TL)^2A_1$, where $A_1\in\mathbb{R}^{m\times r}$ is a full rank matrix, where $r\leq\min\{m,n\}$. I know that $rank(Y_2)=r$ and its QR decomposition is given by $$Y_2=QR=[Q_r\;\; \widetilde{Q}_r]\left[\begin{array}{c}R_r\\0\end{array}\right],$$ where $R_r\in\mathbb{R}^{r\times r}$ is a upper triangular matrix. Also, I know that $\widetilde{Q}_r^TY_2={\bf 0}$, i.e., $$\widetilde{Q}_r^T(L^TL)^2A_1=\widetilde{Q}_r^TL^T[L(L^TL)]A_1={\bf 0}.$$ How can I prove that $\widetilde{Q}_r^TL^T={\bf 0}?$. Any suggestions will be of great help.