diff --git a/main.tex b/main.tex
index 33b21a053f420db6b65470028eb9afa00e7fe92a..87e022c9835c0740dc568c1918bf267c45c37030 100644
--- a/main.tex
+++ b/main.tex
@@ -181,7 +181,7 @@
 	\end{enumerate}
 	\colorbox{red}{\ref{it:pos}-\ref{it:geq} is false since $a_{1114}^{(10)}<0$ -- needs to be double checked!}
 	I figured this out by observing that $R^{(10)}(p)$ has a pole inside the disk of radius $0.96$. This also means that $R^{(10)}(p)=\sum_{k=0}^{\infty}a_k^{(10)}p^k$ is only true in an analytic sense, since for $p>0.96$ the right hand side does not converge.
-	
+
 	We also conjecture that $p_c\approx0.61$, see Figure~\ref{fig:coeffs_conv_radius}.
 
 	\begin{figure}[!htb]\centering
@@ -191,6 +191,31 @@
 	\label{fig:coeffs_conv_radius}
 	\end{figure}
     
+    \newpage
+    \textbf{New conjecture(s)}: The following statements are equivalent
+    \begin{enumerate}
+        \item $p<p_c$
+        \item (requires continuous time) $\P^{[0,\infty]}(\NZ{0}) > 0$
+        \item (requires continuous time) $\P^{[-\infty,\infty]}(\NZ{0}) > 0$
+        \item (requires continuous time) $\exists c,\lambda$ such that $\P^{[-\infty,\infty]}(\Z{[0,k]}) \leq c \; e^{-\lambda \; k}$
+        \item (requires continuous time) $\exists c,\lambda$ such that $\mathrm{COV}^{[-\infty,\infty]}(A,B) \leq c \; e^{-\lambda \; d(A,B)}$
+        \item (requires continuous time) $\E^{(\infty)}(\text{\# resamples of }0) < \infty$
+        \item (first lines requires continuous time)
+            \begin{align*}
+                \P^{[-\infty,\infty]}(\textsc{NonStop})
+                &= \P^{[-\infty,\infty]}(\textsc{NonStop} \text{ and unbounded}) \\
+                &= \P^{[-\infty,\infty]}(\bigcap_{n=1}^{\infty} \Z{[-n,n]}) \\
+                &= \lim_{n\to\infty} \P^{[-\infty,\infty]}(\Z{[-n,n]}) \\
+                &\overset{?}{=} \lim_{n\to\infty} \P^{[-n,n]}(\Z{[-n,n]})
+            \end{align*}
+        \item (does \emph{not} requires continuous time) $\P^{[-\infty,\infty]}(\textsc{NonStop} \mid \text{start with single 0 at 0})$
+        \item $\lim_{n\to\infty} \P^{[0,n]}(\Z{n} \mid \text{start with single 0 at 0}) = 0$
+        \item $\lim_{n\to\infty} \P^{[0,n]}(\NZ{0}) > 0$ (note: symbolically computable for small n)
+        \item $\lim_{n\to\infty} \P^{[0,n]}(\Z{0}) < 1$ (note: symbolically computable for small n)
+        \item For the non-terminating process (resample a random 1 in case there are only 1s available), the number of zeroes in the stationary distribution is $o(n)$.
+    \end{enumerate}
+
+    \newpage
     For reference, we also explicitly give formulas for $R^{(n)}(p)$ for small $n$. We also give them in terms of $q=1-p$ because they sometimes look nicer that way.
     \begin{align*}
     	R^{(3)}(p) &= \frac{1-(1-p)^3}{3(1-p)^3}