Lawler, Schramm and Werner conjectured that the number $R_n$ of self-avoiding walks in the rectangle $[-N;N]\times[0;N]$ connecting the origin to the point $(0;N)$ behaves like
$$
R_n\sim c\cdot N^{-2a}\mu^N,
$$
where $a=5/8$ and $\mu$ is a lattice-dependent quantity known as the connective constant (it is known to be $\sqrt{2+\sqrt{2}}$ for hexagonal lattice and only numerically known for other lattices). Moreover, they conjectured that the measure should have a conformally covariant scaling limit, meaning that if $\Omega$ is another simply-connected domain and $x,y\in \partial \Omega$, with the boundary being nice (say, horizontal or vertical straight lines) near $x,y$, then the number of SAW from $a$ to $b$ in a $\delta$-mesh lattice approximation to $\Omega$ should behave like
$$
R^\Omega_n \sim c |\varphi'(x)|^{a}|\varphi'(y)|^{a}\delta^{2a}\mu^{\delta^{-1}},
$$
where $\varphi$ is a conformal map from $\Omega$ to the rectangle $(-1,1)\times(0,1)$ that maps $x$ to the origin and $y$ to $i$. (We can also fix $|\varphi'(x)|=1$, and then $|\varphi'(y)|$ is proportional to the normal derivative of the Poisson kernel with a pole at $x$.) A straightforward extension of their conjecture would be that the number of SAW from the origin to $(N,b)$, with a given $b$, behaves like
$$
P'_n\sim c(\theta)^a\cdot N^{-3a}\mu^N,
$$
where $c(\theta)$ is proportional to the normal derivative at $(1;\theta)$ of the Poisson kernel in $(0;1)^2$ with a "pole" at the origin, and we are in the regime $b/N\to \theta\in (0,1)$. (If $b\equiv N$ or $b\equiv 0$, the power law exponent should change to $-4a$.)

No-one doubts the conformal covariance ansatz of Lawler, Schramm and Werner; but it is wide open to prove it rigorously. The best you can do rigorously is, probably,
$$
\log P'_n\sim n\log\mu.
$$
I am not sure this is explicitly done anywhere, but the general idea is that all 'reasonable' models of planar SAW should obey this property with the same $\mu$; for instance, here it is done for arbitrary paths, bridges, and closed paths.