Universal enveloping algebra expression for the Burnside transition matrices

Let $e,f,h$ be the basis elements of $\mathfrak{sl}_2$, and define the $2\times 2$ matrices

$$p^+ = \frac{1}{2}(1+e+f),\qquad p^- = \frac{1}{2}(1-e-f),\qquad p^+h = \frac{1}{2}(1+e+f)h.$$

For nonnegative integers $x,y,z$, let $f(x,y,z)$ be the sum over all orders of taking the matrix tensor product of $x$ copies of $p^+$, $y$ copies of $p^-$, and $z$ copies of $p^+h$. The universal enveloping algebra conjecture. The transition matrix $K_n$ satisfies

$$K_n = \sum_{x+y+z=n} c_{y,z} f(x,y,z),$$

where

$$c_{y,z}=\begin{cases}\left(\frac{y!z!}{\left(\frac{y}{2}\right)!\left(\frac{z}{2}\right)!\left(\frac{y+z}{2}\right)!2^{y+z}}\right)^2 & \text{if } y,z \text{ are nonnegative even integers},\\ 0 & \text{otherwise}.\end{cases}$$

This identity has been checked up to $n=10$ and is proposed as an algebraic explanation for the relations among the matrices $K_n$ and their nonzero eigenvalues.