Parallelization constructs in OpenMP, such as parallel for or taskloop, are typically restricted to loops that have no loop-carried dependencies (DOALL) or that contain well-known structured dependence patterns (e.g. reduction). These restrictions prevent the parallelization of many computational intensive may DOACROSS loops. In such loops, the compiler cannot prove that the loop is free of loop-carried dependencies, although they may not exist at runtime. This paper proposes a new clause for taskloop that enables speculative parallelization of may DOACROSS loops: the tls clause. We also present an initial evaluation that reveals that: (a) for certain loops, slowdowns using DOACROSS techniques can be transformed in speed-ups of up to 2.14× by applying speculative parallelization of tasks; and (b) the scheduling of tasks implemented in the Intel OpenMP runtime exacerbates the ratio of order inversion aborts after applying the taskloop-tls parallelization to a loop.