That question about more K was raised in this discussion about late season fertilizer application. And the answer is, when looking for the beneficial effects of potassium (K), those effects are seen only when a deficiency is corrected.

Adding more K to turf sufficient in K does nothing. Adding more K to turfgrass that is deficient in K will promote stronger cells and healthier plants and all the benefits that are commonly associated with K. But this effect disappears once the grass has been supplied with enough K. When the grass has enough, adding more K will have no effect. I recommend these four papers for a better understanding of this:

• A hypothesis relating critical potassium concentrations for growth to the distribution and functions of this ion in the plant cell, by Leigh and Wyn Jones
• Evidence, regulation, and consequences of nitrogen-driven nutrient demand by turfgrass, by Kussow et al.
• Does potassium fertilizer really increase roots?, by Woods
• Turfgrass nutrient guidelines, peer review, and potassium, by Woods

Cool-season grasses use nitrogen and potassium in about a 2:1 ratio. With warm-season grasses, it is closer to 3:2 (for Cynodon and Zoysia) or 1:1 (for Paspalum vaginatum). Applying that much K almost guarantees that the grass will be supplied with enough. Adding more K beyond the amount the grass can use doesn't provide a benefit, and does not promote stronger cells.

In fact, one can often apply as fertilizer less K than the grass will use. This is possible because the grass can make use of potassium in the soil. One needs to test the soil to learn just how much potassium can be supplied from the soil, and how much must be added as fertilizer. This GCM article explains that approach.