Monotonicity conjecture for the coupled Young graph dimension array

Let $K(n,k,l)$ be the recursively defined array of integers associated with the pascalized coupled Young graph, with $n\geq 3$, $k,l$ satisfying $2k+l<n-2$ and $2k+l\equiv n\pmod 2$. Monotonicity conjecture for $K$. One has

$$\frac{K(n-2,k,l)}{K(n,k,l)}\geq\max\left(\frac{K(n-2,k+1,l)}{K(n,k+1,l)},\frac{K(n-2,k,l+2)}{K(n,k,l+2)}\right).$$

In particular,

$$\max_{\substack{2k+l<n\\2k+l\equiv n\pmod 2}}\frac{K(n-2,k,l)}{K(n,k,l)}=\frac{K(n-2,0,\delta(n))}{K(n,0,\delta(n))},$$

where $\delta(n)=0$ if $n$ is even and $\delta(n)=1$ if $n$ is odd. The paper reports verification for many cases but no proof, and presents this numerical claim as implying the preceding boundary-support conjecture.