That would depend on time of year & region of the country, as well as size/depth of that particular lake.
As evidenced by this graph - there is no real difference in temp at depth "at that date" & the lake is Green River Lake in south central KY.
While this chart of Taylorsville Lake, located in west central KY, shows a similar trend, but is a much smaller lake than Green River Lake.
Both of these lakes are normally "lightly stained to stained", so a "very stained to murky" water lake or a "generally clear to lightly stained" lake may have a somewhat different temperature range at depth. I don't have access to charts for those lakes in KY, because they are not COE lakes of the Louisville Region, which is where I got the charts above.