Okay,from here,as an end-user,0293 - Eigo ga Nigate na Otona no DS Training SEEMS to boot,run,and save just fine.
As i said,i cannot really understand it,but i can make some character markings with the stylus,keep noodling for a while,turn off ds.Turn on ds,try to boot a game from mk3.Asks-save:0293 - Eigo ga Nigate na Otona no DS,yes,load other game save,yes,boot other game,yes.Turn off ds.turn on ds,boot 0293 - Eigo ga Nigate na Otona no DS Training,save other game save,yes,load 0293 - Eigo ga Nigate na Otona no DS .sav,yes.game boots,and character markings are still there.
Dont know if this helps.working on 0342 - Doraemon - Nobita no Kyouryuu 2006 DS (J) now.
EDIT-i guess that tells me anything rub rabbits would,so im probably not going to check it now.my son was playing it the other day(ahoy!)but im not sure if he tried to save.