A random graph model on a host graph (Formula presented.) is said to be 1-independent if for every pair of vertex-disjoint subsets (Formula presented.) of (Formula presented.), the state of edges (absent or present) in (Formula presented.) is independent of the state of edges in (Formula presented.). For an infinite connected graph (Formula presented.), the 1-independent critical percolation probability (Formula presented.) is the infimum of the (Formula presented.) such that every 1-independent random graph model on (Formula presented.) in which each edge is present with probability at least (Formula presented.) almost surely contains an infinite connected component. Balister and Bollobás observed in 2012 that (Formula presented.) tends to a limit in (Formula presented.) as (Formula presented.), and they asked for the value of this limit. We make progress on a related problem by showing that (Formula presented.) In fact, we show that the equality above remains true if the sequence of complete graphs (Formula presented.) is replaced by a sequence of weakly pseudorandom graphs on (Formula presented.) vertices with average degree (Formula presented.). We conjecture the answer to Balister and Bollobás's question is also (Formula presented.).