Nontotal Points in Cartesian Restriction Category with Zeroes

Question

I am currently working with Restriction Categories, a kind of categories of partial maps proposed by Cockett and Lack in Restriction Categories I. In Restriction Categories III a cartesian structure on Restriction Categories is considered.*

A restriction terminal object is an object $1$ such that for each object $X$ there is a unique total morphism $!_X$ and for each morphism $f:X \rightarrow Y$ it holds that $!_Yf \leq !_X$, ie. $!_X \overline{f} = !_Yf$. A point in $X$ is, as usual, a morphism $p:1 \rightarrow X$. A restriction zero morphism $0_{XY}:X \rightarrow Y$, which is minimal with respect to the restriction structure.

I want to know, whether it is true that in a cartesian restriction category with zeroes every point other than $0_{XY}$ needs to be total. If this does not hold in general, I am very keen to learn about minimal requirements making it true.

In the archetypical restriction category $\mathsf{Par}$ of sets and partial functions the restriction terminal is the set $1 = \{*\}$ and we have the homset $\mathsf{Par}(1,1) = \{\emptyset, id\}$. This implies that every nonzero point is a total point. In fact, as a restriction terminal object is a terminal object in the subcategory of total maps, every example I know satisfies this.

My motivation for this problem are the following questions.

1. Is it true that any two distinct total points $p,q:X \rightarrow Y$ are disjoint in the sense that there is a complemented restriction idempotent $e$ on $X$ satisfying $ep = p$ and $e^cq = q$?
2. Is it true that any two distinct total points $p,q:X \rightarrow Y$ are disjoint in the sense that they don't have a nontrivial meet, ie. $p \cap q = 0_{1X}$?
3. Is it true that given a total point $p:1 \rightarrow X$ and a complemented restriction idempotent $e$ on $X$ the equation $ep = p$ implies $e^cp = 0_{1X}$?.
4. Conversely, is it true that for a total point $p:1 \rightarrow X$ and a complemented restriction idempotent $e$ on $X$ it always holds that either $ep = p$ or $e^cp = p$?

The examples suggest that some of this should hold true, however I could not derive any of it.

Thank you for your time.

*I think my question is quite involved. I admit that this question might not be a part of mainstream category theory and thus don't really expect to get an answer. Nevertheless I think it may be worth giving it a shot.