Yes and no. Existing crabgrass won't survive frost, so that's the limit there. In Zone 5, frost isn't that far off.

Those seeds? Decades of survival time, theoretically (years is a certainty). Removal of the lasagna in June is just in time for those nice seeds to sprout as soil temperatures rocket above 70ð, and crabgrass is happy to sprout any time between then and when it drops under 70ð again in fall.

Use of sufficient corn gluten meal (not corn, this stuff is bright yellow) can organically suppress out seed sprouting in crabgrass. Reports on efficacy differ wildly and I can't find any rhyme or reason to point me in a direction that would assure better performance.

Other than that, chemical pre-emergents like Barricade do work, but definitely aren't organic. Killing it out with a good crabgrass killer when it appears, ditto.

You can spot-treat with horticultural vinegar in hot, dry, very sunny weather. That's reasonably effective, but the vinegar will kill anything it gets on. Also, wear eye protection as it's acidic enough to damage the corneas of your eyes if it gets in them.