Linear tangency conjecture for pairwise intersecting bi-infinite x-monotone curves

Let $k$ be a fixed positive integer, and let a family consist of $n$ pairwise intersecting bi-infinite $x$-monotone curves such that any two curves intersect at most $k$ times. Linear tangency conjecture. The family admits $O_k(n)$ tangencies. This would extend the corresponding linear bound for $1$-intersecting curves to every fixed intersection parameter $k$; the statement is presented as a suggested generalization and its resolution is not specified here.