Identifiability conjecture for causal models with $H_1$ entropy

Consider the causal model $\mathcal{M}=(\{X,Y\},E,f,X\rightarrow Y,p_{X,E})$, where the discrete random variables $X$ and $Y$ have $n$ states, $E$ is independent of $X$ and has $\theta$ states. The distribution of $X$ is uniformly randomly selected from the $(n-1)$-dimensional simplex, the distribution of $E$ is uniformly selected from the probability distributions satisfying

$$H_1(E)\leq \log n+\mathcal{O}(1),$$

and $f$ is randomly selected from all functions $f:[n]\times[\theta]\rightarrow[n]$. Identifiability conjecture. With high probability, any $\widetilde{E}$ independent of $Y$ such that

$$X=g(Y,\widetilde{E})$$

for some deterministic function $g$ entails

$$H(X)+H(E)<H(Y)+H(\widetilde{E}).$$

This conjecture proposes that, for randomly selected causal models with low $H_1$ entropy in the environment, the causal direction can be identified by preferring the representation with lower total entropy. The parser provides no evidence resolving the conjecture; its status is therefore open.