Classification conjecture for cubic power functions with optimal second-order differential uniformity

Let $n\geq 4$, let $f:\mathbb{F}_{2^n}\to\mathbb{F}_{2^n}$ be the power function $f(x)=x^d$, and identify exponents under the affine-equivalence transformation $d\mapsto 2^i d$ modulo $2^n-1$. The function is cubic when its algebraic degree is $3$, and optimal second-order differential uniformity means that its second-order differential uniformity attains the optimal value $4$. Classification conjecture. The function $f$ has optimal second-order differential uniformity if and only if

$$d\equiv 2^i(2^{2k}+2^k+1)\pmod{2^n-1}$$

for some integers $k$ and $i$ satisfying $\gcd(k,n)=1$. This is supported by the complete computations reported for $4\leq n\leq 20$ and generalizes the sufficient family proved in the paper; the classification remains open.