The chromatic unit-map conjecture for Johnson–Wilson spectra

Let $BP\langle n\rangle$ denote the Johnson–Wilson spectrum at the prime $2$, with coefficient ring $\mathbb{Z}_{(2)}[v_1,\dots,v_n]$. Assume that $BP\langle n\rangle$ admits the structure of an $E_\infty$ ring spectrum, so that $gl_1 BP\langle n\rangle$ is defined. The chromatic unit-map conjecture. There are maps of spectra

$$\psi_n:\Sigma^{2^{n+1}-2}BP\langle n-1\rangle\longrightarrow gl_1 BP\langle n\rangle$$

whose induced map in homotopy is $x\mapsto xv_n$. More precisely, for every $k\geq 2^{n+1}-2$, $(\psi_n)_*$ is the composite

$$\mathbb{Z}_{(2)}[v_1,\dots,v_{n-1}]_{k-2^{n+1}+2}\hookrightarrow \mathbb{Z}_{(2)}[v_1,\dots,v_n]_{k-2^{n+1}+2}\xrightarrow{\,v_n\,}\mathbb{Z}_{(2)}[v_1,\dots,v_n]_k.$$

The assertion is known under the stated assumptions for $n=1$ and $n=2$; the general case is presented conditionally on the existence of the required $E_\infty$ structure.