The differential multiplicity conjecture for nice exponents

Let $L$ be a finite field of odd characteristic, and let $s$ be an exponent. Write $s\not\approx 1$ for the stated exponent equivalence relation, and call $s$ nice when it has the property defined in the source's preceding discussion. A differential multiplicity is the number of solutions associated with a differential of the power mapping. The differential multiplicity conjecture for nice exponents. If $s\not\approx 1$ and $s$ is nice, then $2$ is a differential multiplicity. The conjecture is based on numerical evidence for odd characteristics from $3$ through $31$; the supplied text does not state a resolution.