Blondeau et al.'s differential 8-uniformity conjecture for two power functions

Let $n=2m$ with $m$ odd, and let $F_d:x\mapsto x^d$ be a power permutation polynomial over $\mathbb{F}_{2^n}$, where

$$d=2^m+2^{(m+1)/2}+1$$

or

$$d=2^{m+1}+3.$$

Blondeau et al.'s conjecture. For each of these two values of $d$, $F_d$ is differentially $8$-uniform, and all values $0,2,4,6,8$ occur in its differential spectrum. The paper presents this as an open conjecture of Blondeau et al.; the supplied text does not indicate a resolution.