The symmetric Gaussian vs. MZ derivatives conjecture

For $n\geq2$, let $D_{\mathcal{A}}^s$ be an $n$-th symmetric generalized Riemann differentiation. A symmetric MZ differentiation is one for which, for every function $f$, both the $(n-2)$-nd symmetric Peano derivative $f_{(n-2)}^s(x)$ and $D_{\mathcal{A}}^sf(x)$ exist if and only if the $n$-th symmetric Peano derivative $f_{(n)}^s(x)$ exists. An $n$-th symmetric Gaussian differentiation is the symmetric generalized Riemann differentiation with Gaussian nodes parametrized by $q\neq0,\pm1$. The symmetric Gaussian vs. MZ derivatives conjecture. In orders at least $3$, each symmetric MZ differentiation is a symmetric Gaussian differentiation. The conjecture is false for $n=1,2$, true for $n=3,4$, and remains open for $n\geq5$.