Chen–Hermite balancing conjecture for Gaussian single increments

Let $(X_t)_{t\in[0,T]}$ be a $1$-dimensional continuous stochastic process whose single increments are Gaussian. Assume that, for every $0\leq s\leq u\leq t\leq T$ and every $n\geq2$,

$$\sum_{i=1}^{n-1}\mathbb{E}[H_i(X_{s,u},-\tfrac{1}{2}(u-s))\cdot H_{n-i}(X_{u,t},-\tfrac{1}{2}(t-u))]=0.$$

Chen–Hermite Gaussianity conjecture. Then $(X_t)_{t\in[0,T]}$ must be a Gaussian process. This would establish joint Gaussianity from Gaussian single-increment laws and the Chen–Hermite balancing identities; the corresponding implication is the missing step in the paper's characterization, while it is immediate under an additional joint-Gaussianity assumption.