The Core Idea

Here's the breakthrough: if $dp[j_1] \geq dp[j_2]$ and $j_1 < j_2$, you never need $j_1$ as a candidate for minimum. Why? Because $j_2$ gives a better(or equal) value AND stays valid longer in the sliding window. When $j_1$ expires, $j_2$ is still there with a better(or equal) answer.

This means you can discard candidates that will never be optimal. You only keep the ones that might matter. The monotonic queue does this automatically by removing dominated elements.