Conservation conjecture for truncated modified Hamiltonians in symplectic Euler

Let $\mathcal{P},\mathcal{Q}\subseteq\mathbb{R}^{d}$ be closed and convex, and let $F:\mathcal{P}\rightarrow\mathbb{R}$ and $G:\mathcal{Q}\rightarrow\mathbb{R}$ be $L$-smooth of orders $1,\dots,N+2$. For symplectic Euler with stepsize $\eta>0$, write $\widetilde{H}_{\eta}^{(N)}$ for the $N$th-order truncated modified Hamiltonian and let $z_k$ denote the state after $k$ iterations. Conservation conjecture. For each $N\in\mathbb{N}_{0}$, there exists a monotonic increasing, bounded function $\Phi:\mathbb{N}_{0}\rightarrow\mathbb{Q}$ such that

$$\left|\widetilde{H}_{\eta}^{(N)}(z_{k})-\widetilde{H}_{\eta}^{(N)}(z_0)\right|\leq k\Phi(N)L^{N+3}\eta^{N+2}.$$

This conjecture predicts that the conservation error grows linearly with the number of iterations, improves with the truncation order, and decreases with the stepsize. The source presents it as an expected but unproved dependence on the problem parameters.