The exact piercing-number conjecture for the chessboard

Let $Q_n$ be the $n\times n$ chessboard, and let $p_n$ denote the minimum number of lines whose union intersects every cell of $Q_n$. Piercing-number conjecture. For all $n \geq 3$,

$$p_n=n-1.$$

The conjecture would determine the exact piercing number of the chessboard. The preceding results establish the upper bound $p_n\leq n-1$ for $n\geq 3$ and the asymptotic lower bound $p_n>0.7n$ for sufficiently large $n$; the claimed equality remains open, motivated in the source by a computer search finding no configuration of $n-2$ lines for $n\leq 15$.