Broke it down by decade to see who has had the most consistent success.
Before 1939 (first year of NCAA Tournament)
1) KU 2 helms, 20x conference champions, 483 wins
2) UK 1 helms, 7x conf champs, 326 wins
3) UNC 1 helms, 6x conf champs, 386 wins
4) UCLA 0 helms, 6x conf champs, 187 wins
5) Indiana 0 helms, 3x conf champs, 344 wins
6) Duke 0 helms, 0 conf champs, 356 wins
1939-1949 (1939 had zero blue bloods and none of us won the conference)
1) UK 2 titles, 3 FF's, 4 appearances, 6x conf champs, 255 wins
2) Indiana 1 title, 1 FF, 1 appear, 0 conf champs, 156 wins
3) KU 0 titles, 1 FF, 2 appear, 5x conf champs, 168 wins
4) UNC 0 titles, 1 FF, 2 appear, 3x conf champs, 206 wins
5) Duke 0 titles, 0 appear, 3x conf champs, 181 wins
6) UCLA 0 titles, 0 appear, 1x conf champs, 122 wins
1950-1959
1) UK 2 titles, 2 FF's, 7 appearances, 7x conf champs, 229 wins
2) KU 1 title, 3 FF's, 3 appear, 5 conf champs, 171 wins
3) Indiana 1 title, 1 FF, 3 appear, 4 conf champs, 154 wins
4) UNC 1 title, 1 FF, 2 appear, 3 conf champs, 168 wins
5) UCLA 0 titles, 0 FF's, 3 appear, 3 conf champs, 193 wins
6) Duke 0 titles, 0 FF's, 1 appear, 2 conf champs, 180 wins
1960-1969
1) UCLA 5 titles, 6 FF's, 7 appearances, 7x conf champs, 234 wins
2) Duke 0 titles, 3 FF's, 4 appear, 4x conf champs, 213 wins
3) UNC 0 titles, 3 FF's, 3 appear, 5x conf champs, 184 wins
4) UK 0 titles, 1 FF, 6 appear, 5x conf champs, 204 wins
5) KU 0 titles, 0 FF, 3 appear, 3x conf champs, 173 wins
6) Indiana 0 titles, 0 FF's, 1x conf champs, 135 wins
1970-1979
1) UCLA 5 titles, 7 FF's, 10 appearances, 10x conf champs, 272 wins
2) UK 1 title, 2 FF's, 7 appear, 7 conf champs, 223 wins
3) Indiana 1 title, 2 FF's, 4 appear, 4x conf champs, 208 wins
4) UNC 0 titles, 2 FF's, 6 appear, 6x conf champs, 239 wins
5) KU 0 titles, 2 FF's, 4 appear, 4x conf champs, 178 wins
6) Duke 0 titles, 1 FF, 2 appear, 1x conf champs, 162 wins
1980-1989
1) Indiana 2 titles, 2 FF's, 9 appearances, 5x conf champs, 228 wins
2) UNC 1 titles, 2 FF's, 10 appear, 6x conf champs, 281 wins
3) KU 1 titles, 2 FF's, 6 appear, 1x conf champs, 219 wins
4) Duke 0 titles, 3 FF's, 7 appear, 1 conf champs, 226 wins
5) UK 0 titles, 1 FF, 9 appear, 6x conf champs, 231 wins
6) UCLA 0 titles, 1 FF, 5 appear, 2x conf champs, 201 wins
1990-99
1) Duke 2 titles, 5 FF's, 9 appearances, 6x conf champs, 271 wins
2) UK 2 titles, 4 FF's, 8 appear, 5x conf champs, 282 wins
3) UNC 1 title, 5 FF's, 10 appear, 2x conf champs, 270 wins
4) UCLA 1 title, 1 FF, 10 appear, 4x conf champs, 240 wins
5) KU 0 titles, 2 FF's, 10 appear, 7x conf champs, 286 wins
6) Indiana 0 titles, 1 FF, 10 appear, 2x conf champs, 229 wins
2000-10
1) UNC 2 titles, 4 FF's, 8 appearances, 5x conf champs, 251 wins
2) KU 1 title, 3 FF's, 10 appear, 7x conf champs, 282 wins
3) Duke 1 title, 2 FF's, 10 appear, 4x conf champs, 291 wins
4) UCLA 0 titles, 3 FF's, 8 appear, 3x conf champs, 227 wins
5) UK 0 titles, 0 FF, 9 appear, 4x conf champs, 240 wins
6) Indiana 0 titles, 1 FF, 7 appear, 1x conf champs, 187 wins
2010-2019
1) Duke 2 titles, 2 FF, 10 appearances, 1x conf champs, 299 wins
2) UK 1 title, 4 FF's, 9 appear, 5x conf champs, 305 wins
3) UNC 1 title, 2 FF, 9 appear, 5x conf champs, 277 wins
4) KU 0 titles, 2 FF, 10 appear, 9x conf champs, 304 wins
5) UCLA 0 titles, 0 FF, 6 appear, 1x conf champs, 215 wins
6) Indiana 0 title, 0 FF, 4 appear, 2x conf champs, 195 wins
2020-22
1) KU 1 title, 1 FF, 2 appear, 2x conf champs, 83 wins
2) UNC 0 titles, 1 FF, 2 appear, 2x conf champs, 61 wins
3) UCLA 0 titles, 1 FF, 2 appear, 0x conf champs, 68 wins
4) Duke 0 titles, 1 FF, 1 appear, 1x conf champs, 70 wins
5) UK 0 titles, 0 FF, 1 appear, 1x conf champs, 60 wins
6) Indiana 0 titles, 0 FF, 1 appear, 0x conf champs, 53 wins
Average
2.9 Kentucky
2.9 North Carolina
3.1 Kansas
3.8 Duke
3.9 UCLA
4.4 Indiana