Matrix inequality conjecture for zero diagonal entries

Let $M$ be a matrix with non-negative real entries, let $t$ be an odd positive integer, and let $(x_1,y_1),\ldots,(x_t,y_t)$ be distinct entries of $M$ satisfying $M_{x_i y_i}=0$ for every $i=1,\ldots,t$. Matrix inequality conjecture. Then

$$\sum_{i=1}^{t}\sum_{j=1}^{t}M_{x_i y_j}M_{x_j y_i}\leqslant (t-1)\lVert M\rVert_{\infty}\lVert M\rVert_{1}.$$

The paper formulates this inequality because it implies the weak balls conjecture and proves it in a special case; its general validity remains open.