Convergence conjecture for the modified sphere-packing auxiliary functions

Let $f_k$ and $\widehat f_k$ be the auxiliary functions obtained by forcing the modified roots described in the paper, viewed as analytic functions of a radial variable. Convergence conjecture. As $k\to\infty$, $f_k$ converges to a function $f$ and $\widehat f_k$ converges to $\widehat f$ on some neighborhood of the real line in $\mathbb C$, with convergence uniform on compact subsets of that neighborhood. Such convergence would produce analytic, rapidly decreasing limiting functions and hence admissible auxiliary functions; the source reports numerical evidence but no proof.