The complementary Bell numbers or Uppuluri–Carpenter numbers, denoted $\tilde{B}_n$, can be delivered by $$G(x):=\sum_{n\geq0}\tilde{B}_n\frac{x^n}{n!}=e^{1-e^x}.$$ Definition. Fix an integer $m\geq0$. We say $F(x)=\sum_{n\geq0}a_n\frac{x^n}{n!}$ has the index-m property whenever $a_n=0$ iff $n=m$ is satisfied.
In this language, Herb Wilf's conjecture can be stated as: $G(x)$ has the index-2 property.
By now, there are a few papers turning this into a theorem, including this one:
Tewodros Amdeberhan, Valerio De Angelis and Victor H. Moll, Complementary Bell Numbers: Arithmetical Properties and Wilf’s Conjecture (2013) In: Kotsireas I., Zima E. (eds) Advances in Combinatorics. Springer, Berlin, Heidelberg, doi:10.1007/978-3-642-30979-3_2 (pdf)
The other day, I started pondering on a truncated version of the generating function, namely, for $\pmb{r\geq2}$, define $$G_r(x)=e^{-(x+\frac{x^2}{2!}+\frac{x^3}{3!}+\dots+\frac{x^r}{r!})}=\sum_{n\geq0}\tilde{B}_{n,r}\frac{x^n}{n!}.$$ Of course, $\lim_{r\rightarrow\infty}G_r(x)=G(x)$. After some experimentation, I'm enough convinced to ask:
Question 1. Fix $r\geq2$. Is it true that $G_r(x)$ has the index-2 property?
Question 2. Does Wilf's conjecture imply the claim in Question 1? Or, vise-versa?
An aside: the functions $G_r(x)$ are equally interesting as related to partition theory, Hermite polynomials, permutations, Young tableaux, etc.
UPDATE.
(1) From $G_{r+1}(x)=G_r(x)\cdot e^{-\frac{x^{r+1}}{(r+1)!}}$, it follows that $\tilde{B}_{n,r+1}=\tilde{B}_{n,r}$ for all $n\leq r$. By the same reasoning, $\tilde{B}_n=\tilde{B}_{n,r}$ for all $n\leq r$. It is now clear that $\tilde{B}_{n,r}=0$ iff $n=2$, for all $r$, implies $\tilde{B}_n=0$ iff $n=2$. This related to a part of Question 2.
(2) It is also possible to prove if $G(x)$ has index-2 property then so does $G_r(x)$, for large enough $r$. Again, this is related to a part of Question 2.
(3) The case $G_2(x)$ has index-2 property is known:
Solution by R. Israel to a problem proposed by F. Schmidt and R. Simion, Even minus odd involutions in the symmetric group, SIAM Rev. 34 (2), pages 315-317 (1992) doi:10.1137/1034062.
