Denote by $\circ$ the Kronecker product, let $|\cdot|$ denote the matrix/vector of absolute values, and let $e$ be the vector of all ones. Comparison is entrywise, i.e., $y \ge x$ is equivalent to $y_i \ge x_i$ for all $I$.
Problem 1: Let $A$ be a real $n\times n$ matrix, $n>1$, and assume
$$
(A \circ A)e = (n-1)e.
$$ Then there exists $x \neq 0$ with $|Ax| \ge |x|$.
A weaker formulation (Problem 2): Assume $$ |Ae| = n e. $$ Then there exists $x \neq 0$ with $|Ax| \ge |x|$.
The first problem is sharp in the sense that assuming $(A \circ A)e = (n-1)e$, then there exists no $x \neq 0$ with $|Ax| > |x|$.
Equality, i.e., there exists $x \ne 0$ with $|Ax| \ge |x|$, is true for the matrix with zero on, $+1$ above and $-1$ below the diagonal, i.e. for $A=(a_{ij})$ with $$ a_{ij}= \begin{cases} +1 & i<j\\ 0 & i=j\\ -1 & i>j\\ \end{cases} $$
