Linear resolution of products of monomial ideals related to maximal minors

Let $X$ be an $m \times n$ matrix of distinct indeterminates over a field $K$, where $m \le n$. Set the polynomial ring $K[X] := K[X_{ij} : 1 \le i \le m, 1 \le j \le n]$. Let $1 \le k < l \le n$ be such that $l - k + 1 \ge m$. Consider the submatrix $Y_{kl}$ of consecutive columns of $X$ from $k$th column to $l$th column. Let $J_{kl}$ be the ideal generated by `diagonal monomials' of all $m \times m$ submatrices of $Y_{kl}$, where the diagonal monomial of a square matrix means product of its main diagonal entries. We show that $J_{k_1 l_1} J_{k_2 l_2} \cdots J_{k_s l_s}$ has a linear free resolution, where $k_1 \le k_2 \le \cdots \le k_s$ and $l_1 \le l_2 \le \cdots \le l_s$. This result is a variation of a theorem due to Bruns and Conca. Moreover, our proof is self-contained, elementary and combinatorial.