I definitely do not have those numbers, but just looking at Wikipedia since the 2019-20 season, these are MSU's next home results directly following a road loss in Big Ten play:
2019-20
L 71-42 at Purdue --> W 67-55 vs. Wisconsin
L 64-63 at Wisconsin --> L 75-70 vs. #22 Penn State
2020-21
L 84-78 at #8 Iowa --> W 66-56 vs. Nebraska
L 73-55 at Maryland --> W 64-58 vs. Indiana
L 69-50 at #2 Michigan --> W 70-64 vs. #2 Michigan (weird COVID schedule)
2021-22
L 56-55 at #24 Illinois --> W 83-67 vs. Michigan
L 84-63 at Rutgers --> L 70-62 vs. #14 Wisconsin
L 62-58 at Penn State --> L 79-74 vs. #12 Illinois
L 86-60 at #25 Iowa --> W 68-65 vs. #4 Purdue
L 80-69 at #23 Ohio State --> W 77-67 vs. Maryland
2022-23
L 75-66 at Illinois --> L 64-63 vs. #3 Purdue
L 82-69 at Indiana --> W 63-61 vs. Iowa
L 84-72 at Michigan --> W 80-65 vs. #17 Indiana
2023-24
L 71-68 at #10 Illinois --> W 73-55 vs. Rutgers
L 81-66 at #13 Wisconsin --> W 81-62 vs. Michigan
L 59-56 at Minnesota ---> ?????? vs. #10 Illinois
So, not really a pattern.
I guess the most you could say is that playing an average or below average team at home after a Big Ten road loss does look to be a bit of "medicine" to right the ship. However, in games where they returned home to play a ranked team, they were 3-3 ... and 0-1 against the Illini! So, right back to the same story ... if we think we are better than MSU and we come out and play our game, it will not matter how "desperate" they feel and we will walk away with a massive Quad 1 win.