Given some point $x$ on the 2D grid $\mathbb{Z}^2$, is there an upper bound on the number of self avoiding walks of length $n$ between the origin and this point?
I have looked at some literature on the topic but most writing seems to be primarily concerned with the total number of walks of length $n$ regardless of end point. Upper bounds exist on this quantity [1] (often denoted $c_n$) but I was unable to find any prominently displayed results for the case where a destination is specified (the value in question is often denoted $c_n(x)$).
I found a post where someone asked a similar question [2] but the answer only provided a naive lower bound and I'm interested in an upper bound.
[1] https://projecteuclid.org/download/pdfview_1/euclid.ecp/1518426011
[2] Number of simple paths between two vertices on an $n \times m$ square-grid graph?