Private Nursing Homes, Product Differentiation and Reducing COVID-19 Contagion Risk

At dinner tonight, my wife told me that 1/2 of COVID-19 deaths have taken place at nursing homes.
I asked her a Chicago Price Theory question.  I said; “If up to date COVID-19 Nursing home specific death rates are posted, won’t private nursing homes devote more effort to reducing their death rate to better compete for customers?”   The net effect of this competition among differentiated products will be lower COVID-19 contagion risk for the elderly who live in these homes.   The children of the elderly have strong incentives to research the relative risk at these nursing homes and to transfer their elderly parents from the dangerous nursing homes. 

Do public nursing homes have equally strong incentives to minimize contagion risk?  In my co-authored 2017 paper, we discuss the benefits of privatization for reducing public transit costs. In this nursing home case, does privatization help to reduce the negative externality?

Here are the current New York State death counts by nursing home. I realize that a denominator is not included listing the current capacity at each nursing home. 

My point is that nursing homes are differentiated products and competing on low contagion death rate should be another measure of quality.  If those nursing homes with higher death counts lose paying customers then this market discipline encourages risk minimization. Market forces will help to reduce the death count from this disease.