Having quackgrass for many years and it surviving through a renovation, it is safe to say any amount of mowing and mowing heights will not get rid of it. Its rhizomatous roots will always prevail sending up new shoots to tower over your grass.
My advice is the white glove of death method. I've been on squatting through my yard since last fall and 50% glyphosate concentrate, 50% water, and some NIS is the kiss of death.
It is extremely overwhelming, but one bite at a time and you will slowly get it. Stay determined and it will be gone with time.
This or switch your cultivar to ProVista *** and spray Glyphosate to your hearts content.
I'm afraid I'm now at the same conclusion. I mixed 1 part to 3 gly to water and although I'm seeing kill on quack that had the majority of the blades covered, it's not doing much where I was only able to touch the tips of the blades.
A ProVista renovation is probably the only way forward for me. I'm going to try Plan B for the next season or two before I'm willing to do that though, and that's, fertilize and mow like hell.
I also don't think mowing high is going to work at all. The quack just gets tall and then lays down sideways and isn't getting crowded out when it's more vertical. My current game plan has changed to lowering my HOC from ~3.5+ to 1.75 over the coming month so I'll at least bag a bunch of quack. Then I'll dethatch with a SunJoe that will rip out the sideways quack (that worked pretty well last fall.)
While a non selective herbicide will work best, I have had luck suppressing and killing quackgrass with tenacity and then fusilade (at the Bermuda grass removal rates). You have to be careful as the ai in fusilade will kill most grasses (slowly) so you have to be sure to apply a rate that will allow your wanted grass to survive. I have some spots if quack still remaining but waiting to kill it after some young grass is more mature. In the meantime tenacity is suppressing it.
I did a broadcast spray of tenacity last spring and it did whiten the quack. I've also tested Ornamec 170 which is a less concentrated version of fusilade and does indeed kill quack when spot sprayed, but does a number on my KBG.
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@Stock Car Mafia said, please share more on this. What's your turf type?