Let $U$ be an $n\times n$ orthogonal matrix, i.e. $U\in\mathbb{R}^{n \times n}$. For any non-empty ordered sets $S_1,S_2\subset\{1,2,...,n\}$, define $U_{S_1S_2}$ to be an $|S_1|\times|S_2|$ submatrix of $U$ which consists of the intersection entries of rows in $S_1$ and columns in $S_2$. Let $\odot$ be the Hadamard product (element-wise product). Here $|S|$ is the cardinality of set $S$. More precisely, an ordered set of cardinality $k$ can be written as a $k$-vector with distinct entries $(i_1,...,i_k)$. Therefore $$U_{(i_1,...,i_k),(j_1,...,j_k)}=\begin{pmatrix} U_{i_1j_1} & \cdots & U_{i_1j_k}\\ \vdots & \ddots & \vdots\\ U_{i_kj_1} & \cdots & U_{i_kj_k} \end{pmatrix}$$ Then is the following conjecture true?

For any unit vector $v=(v_1,...,v_k)$ in $\mathbb{R}_+^k, k\leq n$ and any orthogonal matrix $U\in O(n)$, there exists ordered subsets $S_1,S_2\subset\{1,2,...,n\}, |S_1|=k,|S_2|=k$, such that: $$\sum_{j=1}^n\left(\sum_{i=1}^kv_iU_{ij}^2\right)^2\leq v^T[U_{S_1S_2}\odot U_{S_1S_2}]v$$

If this is not true, is the following weaker conjecture true? $$\sum_{j=1}^k\left(\sum_{i=1}^kv_iU_{ij}^2\right)^2\leq v^T[U_{S_1S_2}\odot U_{S_1S_2}]v$$

The conjecture can be easily verified when $k=1$. The weaker conjecture can also be verified when $v=(1/\sqrt{k},...,1/\sqrt{k})$.

Maybe we can start from the particular case that $v=(1/\sqrt{k},...,1/\sqrt{k})$.