The shift homomorphism conjecture for divided-power generators

Let $p$ be the characteristic of the field $\mathbb{K}$, let $X=\{e_{i,i+1}^{(p^k)}\mid 1\le i\le n-1,\\ k\in\mathbb{N}_0\}$, and write $a_{ik}=e_{i,i+1}^{(p^k)}$. Equip $X$ with the ordering $a_{11}<a_{21}<\dots<a_{n-1,1}<a_{12}<\dots$ and equip $\mathfrak{U}_{\mathbb{K}}(sl^+_n)$ with deglex ordering, where $\deg(a_{ik})=p^k$. Shift homomorphism conjecture. The map $X\to\mathfrak{U}_{\mathbb{K}}(sl^+_n)$, $a_{ik}\mapsto a_{i,k+j}$, can be extended to a homomorphism of graded algebras

$$\mathfrak{U}_{\mathbb{K}}(sl^+_n)\to\mathfrak{U}_{\mathbb{K}}(sl^+_n).$$

The claim is known in the case $n=3$ and $p=2$, but its status in the generality stated here is not resolved in the source.