Infinite sign-change conjecture for the orthorecursive coefficients

Let $(c_n)_{n\geq0}$ be the real coefficient sequence, and say that it changes sign at $N>0$ when $c_Nc_{N+1}<0$.

Sign-change conjecture. The sequence $(c_n)_{n\geq0}$ changes sign infinitely often.

Numerical experiments find sign changes at $0,1,27,533,10457,\ldots$, but do not establish infinitude; the conjecture concerns the continuation of this observed oscillation.