Linear-length measurement sequence conjecture for fault-tolerant error correction

Let $[n,k,d]$ be a family of linear codes with polynomial distance satisfying

$$d\geq \Omega(n^\alpha)$$

for some constant $\alpha>0$. A fault-tolerant error-correction sequence is a sequence of parity-check measurements that allows fault-tolerant error correction.

Linear-length measurement sequence conjecture. For any such family, there exists a fault-tolerant error-correction sequence consisting of $O(d)$ parity-check measurements.

The paper's upper bound is $O(d\log d)$ measurements, and the conjecture proposes a linear improvement for polynomial-distance codes. The conjecture is presented as unresolved in the source.