Jacobi polynomial zero-preservation conjecture

Let $P_n^{(\alpha,\beta)}(x)$ denote the $n$-th Jacobi polynomial. For a polynomial $f(x)=\sum_{n=0}^N a_nx^n$ whose zeros all lie in $(-1,1)$, define the linear transformation by

$$T_{\alpha,\beta}[f](x)=\sum_{n=0}^N a_nP_n^{(\alpha,\beta)}(x).$$

Jacobi zero-preservation conjecture. If $\alpha$ and $\beta$ are non-negative integers, then $T_{\alpha,\beta}$ preserves the set of polynomials whose zeros lie only in $(-1,1)$. The conjecture is presented as an extension of the proved Legendre case and is reported to have been checked numerically for $\alpha,\beta<5$ in several finite families, but no proof or resolution is supplied.