The kernel conjecture for the second-kind interval transform

Let ${\mathbb Q}\langle a,b\rangle_n$ denote the vector space of homogeneous noncommutative $ab$-polynomials of degree $n$, and let $I_2:{\mathbb Q}\langle a,b\rangle_n\rightarrow {\mathbb Q}\langle a,b\rangle_{n+1}$ be the interval transform of the second kind. Let $A_{\mathbb Q}\langle a,b\rangle_n$ be the specified subspace of ${\mathbb Q}\langle a,b\rangle_n$.

Kernel conjecture. For each $n\geq 1$,

$$\ker I_2=A_{\mathbb Q}\langle a,b\rangle_n.$$

The conjecture identifies precisely the kernel of the second-kind interval transform, whose nontrivial kernel distinguishes it from the corresponding Tchebyshev operator framework. Its status is not resolved in the supplied source.