Binomial-transform conjecture for classical and enhanced k-crossing-avoiding partitions

For $k\geq2$, let $C_k(n)$ be the number of partitions of $[n]$ avoiding classical $k$-crossings, and let $E_k(n)$ be the number avoiding enhanced $k$-crossings. Binomial-transform conjecture. Fix $k\geq2$. The following identity holds:

$$C_k(n+1)=\sum_{i=0}^n{n\choose i}E_k(i).$$

The identity extends the known Catalan–Motzkin relation for $k=2$ and the corresponding identity for $k=3$. A proof is suggested as an open direction via bijections for crossing-avoiding partitions, and the source gives no resolution.