The rules for being Lost in Time and Space says that you are delayed and you lose your next turn:
The investigator is now delayed, and the player should place the investigator marker on its side. The investigator loses his next turn, remaining in the Lost in Time and Space area. The player may only stand his investigator marker back up during the Movement Phase.
Does the fat that the investigator "loses his next turn" and "may only stand his investigator marker back up during the Movement Phase" mean that they skip their Upkeep Phase?