Speculative, parallel abstractions allow that, once a result is computed, the remaining (unnecessary) speculative computations can be safely stopped. However, it is difficult to know when it is safe to stop an ongoing computation. This paper presents a refinement of the parallel speculative ParT abstraction with an affine type system that allows in-place updates, and the killing of speculative computations using thread-local reasoning. There is ongoing work to prove the soundness of the calculus and implement it in the Encore language.