Sarwate's minimum-distance conjecture for binary cyclic codes

Let $m=2t$, let ${\mathbb F}$ be the finite field of size $2^m$, and let ${\mathcal C}_d$ be the binary cyclic code of length $2^m-1$ and dimension $2m$ with two nonzeros ${\alpha}^{-1}$ and ${\alpha}^{-d}$, where $\alpha$ is a primitive element of ${\mathbb F}$ and $\gcd(d,2^m-1)=1$. The Sarwate conjecture. The minimum distance of ${\mathcal C}_d$ is at most

$$2^{m-1}-2^t.$$

This conjecture concerns the minimum distance, equivalently the Walsh spectrum, of binary cyclic codes with two primitive nonzeros. It is attributed to Dilip V. Sarwate and is described in the source as a challenging well-known conjecture; no resolution is supplied here.