E8 and Leech limiting-ratio conjecture

Let $f$ and $\widehat f$ be the limiting functions obtained by forcing roots at lattice vector lengths. For $0<n<10$, use the $E_8$ vector lengths; for $0<n<26$, use the Leech lattice vector lengths. Define

$$p_{24}(n)=n^8-284n^7+35312n^6-2510720n^5+111652352n^4-3180064256n^3+56651266048n^2-577142292480n+2574499479552.$$

E8 and Leech limiting-ratio conjecture. In the $E_8$ case,

$$\frac{f(0)}{\widehat f(0)}=-\frac{n^4-56n^3+1184n^2-11200n+40320}{16(n-10)(n-14)(n-18)}.$$

In the Leech case,

$$\frac{f(0)}{\widehat f(0)}=-\frac{p_{24}(n)}{32(n-26)(n-34)(n-38)(n-42)(n^3-116n^2+4480n-57024)}.$$

These formulas predict the limiting linear-programming ratios across continuous ranges of dimensions and extend the exceptional-dimension investigations beyond $8$ and $24$; the source supplies numerical motivation but no proof.