The White House did once treat Conover with respect. In 1969 it chose him to organize the musical portion of the 70th birthday party that President Nixon gave for Duke Ellington. Willis recruited the all-star band and produced and narrated the concert. 
I took a picture of him that afternoon at the rehearsal in the East Room as he listened to Hank Jones, Gerry Mulligan, Paul Desmond, Clark Terry, Bill Berry, J.J. Johnson, Urbie Green, Jim Hall, Louis Bellson, Milt Hinton, Joe Williams and Mary Mayo. The concert was finally released on a Blue Note CD in 2002. I was honored to write the liner essay. Here’s a bit of it.
Hank Jones, Billy Taylor, and Dave Brubeck played beautifully, but the hands-down winner in the piano category was the 65-year-old Earl Hines, who in two daring minutes of “Perdido†tapped the essence of jazz. Ellington stood up and blew him kisses. Later, Billy Eckstine, who sang with Hines’ band before he had his own, walked up to his old boss and gave him an accolade: “You dirty old man.†The concert lasted an hour and a half, and the room was swinging. I looked around at heads bobbing and shoulders swaying and found Otto Preminger beaming and snapping his fingers Teutonically, one snap at the bottom of each downward stroke of his forearm.
Urged onto the platform, Ellington improvised an instant composition inspired, he said, by “a name, something very gentle and graceful—something like ‘Pat.’†The piece was full of serenity and the wizardry of Ellington’s harmonies. Mrs. Nixon, who looked distracted through much of the evening, paid close attention.
The evening was Ellington’s, gloriously so, but it was Willis’s connections, coordination, organizational skill and stewardship that put the icing on the birthday cake. It was one reason among many larger ones that he deserved, and still deserves, the Presidential Medal of Freedom.