Convex rationalization when the budget sets are segments?
Take a dataset $D = (B^t, x^t)_{t \in T}$ such that for all $t$, $x^t \in B^t$. I'll say that $D$ is rationalisable by the utility function $u$ if for all $t$ and all $x \in B$: $u(x^t) \ge u(x)$.
If you only impose convexity on $u$, then any dataset $D$ is rationalisable by the constant (convex) utility function $u(x) = k$. So this has not testable implications.
Assume now that you impose strict monotonicity on $u$ in the sense that $x &gt; y$ implies $u(x) &gt; u(y)$.
Consider a budget set $\overline{y^t z^t}$ and take the comprehensive hull:
$$
CH(\overline{y^t z^t}) = \left\{x \in \mathbb{R}^n: x \le w \text{ for some } w \in \overline{y^t z^t}\right\}.
$$
Then we can show that the dataset $D = (\overline{y^t z^t}, x^t)_{t \in T}$ is rationalisable by a montone utility function $u$ if and only if the dataset $D' = (CH(\overline{y^t z^t}), x^t)_{t \in T}$ is rationalisable by the same monontone utility function $u$.

For the proof: $(\to)$ Let $D$ be rationalisable by the montone function $u$. We want to show that for all $t$ and all $x \in CH(\overline{y^t z^t})$ we have $u(x^t) \ge u(x)$. Indeed, if $x \in CH(\overline{y^t z^t})$, we know that $x \le w$ for some $w \in \overline{y^t z^t}$. As such:
$$
u(x) \le u(w) \le u(x^t),
$$
where the first inequality comes from the monotonicity conditionon $u$ and the last one from rationalisability of D.


$(\leftarrow)$ Let $D'$ be rationalisable by $u$. Let us first show that for all $t$, $x^t \in \overline{y^t z^t}$. If not, then as $x^t \in CH(\overline{y^t z^t})$ there must exist a $w \in \overline{y^t z^t}$ such that $x^t &lt; w$; Then, by monotonicity of $u$ and rationalisability of $D'$:
$$
u(x^t) &lt; u(w) \le u(x^t),
$$
a contradiction. Now for any $z \in \overline{y^t z^t}$ it then follows from rationalisability of $D'$ that $u(z) \le u(x^t)$, so $D$ is also rationalisable by $u$.

This shows the following equivalence:

The dataset $D = (\overline{y^t, z^t}, x^t)_{t \in T}$ is rationalisable by a convex and montone utility function $u$.
The dataset $D = (CH(\overline{y^t, z^t}), x^t)_{t \in T}$ is rationalisable by a convex and monotone utility function $u$.

As the sets $CH(\overline{y^t z^t})$ are comprehensive, the revealed preference conditions for answering the condition 2) can be found in the JET paper that you refer to (Revealed preference analysis for convex rationalizations on nonlinear budget sets)