The equality-case conjecture for Hansen's theorem

Let $d$ be the parameter in Hansen's theorem and let its lower estimate be the stated inequality involving $d$. The equality-case conjecture. Hansen's theorem continues to hold when the symbol $\geq$ in the lower estimate for $d$ is replaced by $=$.

The conjecture concerns sharpening the lower estimate in Hansen's theorem to an equality case. The excerpt does not reproduce Hansen's theorem or define the parameter $d$, so the precise mathematical content requires checking the source.