Palindromic symmetry conjecture for recurrence coefficients in superelliptic central extensions

Let $A=\mathbb{C}[x^{\pm1},u\mid u^2=P(x)]$, where $P(x)\in\mathbb{C}[x]$ is palindromic of even degree $2r$ with simple roots. Let $A/\partial A$ denote the quotient by exact derivatives, whose reduction relations define recurrence coefficients indexed by degree. Palindromic symmetry conjecture. The identity

$$P(x)=x^{2r}P(1/x)$$

forces the recurrence coefficients in $A/\partial A$ to be symmetric in the degree index. For $r=1$, this symmetry should imply, via Favard's theorem, that the associated polynomial family is orthogonal with respect to a symmetric measure; for $r\geq 2$, the recurrence is expected to be a multi-component system of dimension $r$, requiring a block-Jacobi generalization. The claim is verified in the cases $r=1$ and $r=2$, while no assertion is made that symmetry alone forces the classical Jacobi class. This conjecture proposes a general structural link between palindromic branch-locus symmetry and recurrence relations in universal central extensions; the higher-degree multi-component case remains open, and the stated orthogonality discussion is established on the current-algebra side only for the cited type-$A'$ canonical family.