The fast self-correcting algorithm conjecture for computing π

Let $M(n)$ denote the time complexity of multiplying two $n$-bit integers. The self-correcting algorithm described in the paper computes $pi$ with error $O(2^{-n})$.

Fast self-correcting $pi$ algorithm conjecture. Algorithm~ runs in $O(M(n)\log n)$ bit operations.

The algorithm is already proved to run in $O(M(n)\log^2 n)$ bit operations and is self-correcting, producing a more accurate approximation from an approximate input with cubic convergence. The conjecture asserts the sharper complexity matching the known asymptotically fastest algorithms for computing $pi$; no resolution is supplied in the source.