Conjecture on the roots of the polynomials alpha_l and beta_l

Let $\alpha_l(m)$ and $\beta_l(m)$ be the integer-coefficient polynomials whose existence is asserted by the preceding theorem, characterized by

$$d_l(m)=\frac{1}{l!m!2^{m+l}}\left(\alpha_l(m)\prod_{k=1}^{m}(4k-1)-\beta_l(m)\prod_{k=1}^{m}(4k+1)\right).$$

Root-location conjecture. All the roots of the polynomials $\alpha_l(m)$ and $\beta_l(m)$ lie on the line $\operatorname{Re}(m)=-1/2$. This conjecture is based on extensive numerical calculations; no resolution is supplied here.