Burstein–Han–Kitaev–Zhang shape-Wilf-equivalence conjecture

Let $k\geq 2$. Consider the two partially ordered patterns depicted below: the first has root label $k-1$, lower labels $1,2,k-2,k$ (with the intermediate lower vertices understood from the displayed pattern), and the second has root label $k$, lower labels $1,2,k-1$ (with the intermediate lower vertices understood from the displayed pattern). Write $\sim_s$ for shape-Wilf-equivalence of partially ordered patterns.

Burstein–Han–Kitaev–Zhang conjecture. The two displayed partially ordered patterns are shape-Wilf-equivalent:

$$\text{first displayed pattern}\sim_s\text{second displayed pattern}.$$

This conjecture concerns shape-Wilf-equivalence, a refinement of Wilf-equivalence for partially ordered patterns. The supplied text identifies it as Conjecture 10 proposed by Burstein, Han, Kitaev, and Zhang, but gives no evidence of a resolution.