No-unwanted-roots conjecture for the limiting sphere-packing functions

Let $f$ and $\widehat f$ be the limiting functions from the convergence conjecture, and call the prescribed zeros of the construction the forced roots. No-unwanted-roots conjecture. The limiting functions $f$ and $\widehat f$ have no real roots other than the forced roots. This would prevent unexpected sign changes and, together with convergence, would yield sharp sphere-packing bounds in dimensions $8$ and $24$. The source gives numerical evidence, but the claim remains unproved.