You write that "every scientist ... has a list of unsolved problems he tries to solve". I don't think this is true for every scientist but it is true that some disciplines (e.g., mathematics - see here) maintain lists of problems to be solved.
Similarly, most leading poskim are busy enough learning by themselves, writing and answering the reams of questions reaching them (dozens of questions a day is pretty common, R Shlomo Aviner is said to get 400-600!)
However just as in science, some leading poskim are indeed working on unresolved problems and writing ahead of their times. Two such works I know of (but I'm sure there are more and invite others to edit) are
- R Yechiel Michel Epstein's Aruch HaShulchan he'Atid (Laying the Table of the Future), a parallel work to Aruch HaShulchan summarising and analysing the laws that will apply in Messianic times (from here)
- R Moshe Avigdor Amiel's Ethics and Legality in Jewish Law where the former Chief Rabbi of Tel Aviv writes about how the Torah would apply to a modern state of Israel run fully according to the Torah