Fifth-order differential equation conjecture for the degree-of-symmetry generating function of Dyck paths

Let $D(s,z)$ be the generating function for Dyck paths with respect to their degree of symmetry, with $z$ marking semilength. In the walk formulation, $D(s,z)=R(1,0,s,z)$.

Differential-equation conjecture. The generating function $D(s,z)$ is D-finite in $z$ but not algebraic. Specifically, it satisfies a fifth-order linear differential equation with polynomial coefficients with maximum degree $27$ in $z$.

The conjecture is based on computations by Alin Bostan using the functional equation for the auxiliary walk generating function. The source gives no evidence that it has been proved or disproved.