  We consider the simple random walk on $\mathbb{Z}^d$ killed with probability
$p(|x|)$ at site $x$ for a function $p$ decaying at infinity. Due to recurrence
in dimension $d=2$, the killed random walk (KRW) dies almost surely if $p$ is
positive, while in dimension $d \geq 3$ it is known that the KRW dies almost
surely if and only if $\int_0^{\infty}rp(r)dr = \infty$, under mild technical
assumptions on $p$. In this paper we consider, for any $d \geq 2$, functions
$p$ for which the KRW dies almost surely and we ask ourselves if the KRW
conditioned to survive is well-defined. More precisely, given an exhaustion
$(\Lambda_R)_{R \in \mathbb{N}}$ of $\mathbb{Z}^d$, does the KRW conditioned to
leave $\Lambda_R$ before dying converges in distribution towards a limit which
does not depend on the exhaustion? We first prove that this conditioning is
well-defined for $p(r) = o(r^{-2})$, and that it is not for $p(r) = \min(1,
r^{-\alpha})$ for $\alpha \in (14/9,2)$. This question is connected to
branching random walks and the infinite snake. More precisely, in dimension
$d=4$, the infinite snake is related to the KRW with $p(r) \asymp
(r^2\log(r))^{-1}$, therefore our results imply that the infinite snake
conditioned to avoid the origin in four dimensions is well-defined.
