  The invariant subspace problem (ISP) is a well known unsolved problem in
funtional analysis. While many partial results are known, the general case for
complex, infinite dimensional separable Hilbert spaces is still open. It has
been shown that the problem can be reduced to the case of operators which are
norm limits of nilpotents. One of the most important subcases is the one of
quasinilpotent operators, for which the problem has been extensively studied
for many years. In this paper, we will introduce a new conjecture (supported by
a heuristic argument), and we will prove conditionally that every
quasinilpotent operator has a nontrivial invariant subspace. We will conclude
by posing an open problem which would have deep implications regarding the ISP.
