Open problem on risk-aware planning in the plane

We consider the problem of planning a collision-free path of a robot in the presence of risk zones. The robot is allowed to travel in these zones but is penalized in a super-linear fashion for consecutive accumulative time spent there. We recently suggested a natural cost function that balances path length and risk-exposure time. When no risk zones exists, our problem resorts to computing minimal-length paths which is known to be computationally hard in the number of dimensions. It is well known that in two-dimensions computing minimal-length paths can be done efficiently. Thus, a natural question we pose is "Is our problem computationally hard or not?" If the problem is hard, we wish to find an approximation algorithm to compute a near-optimal path. If not, then a polynomial-time algorithm should be found.