Polynomial-kind uniformity conjecture for factorized $A_2$-Leonard pairs

A factorized $A_2$-Leonard pair consists of compatible parameter arrays

$$\Phi_j=(t_x,t_x^\star(j),x=0,\dots,N-j;f_x(j),g_x(j),x=1,\dots,N-j)$$

and

$$\Psi_x=(\theta_j(x),\theta_j^\star,j=0,\dots,N-x;\varphi_j(x),\phi_j(x),j=1,\dots,N-x),$$

where the arrays are of type II when their eigenvalues are quadratic in the index. A parameter array is associated to a kind of polynomial from the $(q-)$Askey scheme. Polynomial-kind uniformity conjecture. To get a factorized Leonard pair, it is necessary that $\Phi_j$ for any $j$ (respectively, $\Psi_x$ for any $x$) is associated to the same kind of polynomial. The conjecture would strongly restrict the possible combinations of parameter arrays, but the source gives no evidence of a proof or resolution.