Does P(Y|X)=const. in the whole P(Y,X) imply that P(X|Y) also remains constant? Or is covariate shift the same as label shift?

I research covariate shift solutions for ML models. Some papers/books (e.g. "Probabilistic Machine Learning" by Kevin Murphy) claim that one needs different solutions for covariate and label shifts, I claim they are the same.

Definitions:

Covariate shift - P(Y|X) remains the same, P(X) changes between source and target domains.

Label shift - P(X|Y) remains the same, P(Y) changes between source and target domains.

Difference between these two: causality. We speak about covariate shift when X->Y and label shift otherwise.

I claim that if P(Y|X) changes in any point of the feature space X then P(X|Y) changes in the corresponding point of the label space. Therefore, assumption that P(Y|X)=const. implies that P(X|Y)=const. as well in the whole joint P(Y,X).

Am I wrong?

No, $$P(Y\mid X)$$ remaining unchanged over time does not imply that $$P(X\mid Y)$$ also remains unchanged.
$$\begin{pmatrix} 0.2 & 0.1 \\ 0.3 & 0.4 \end{pmatrix} ,\qquad \begin{pmatrix} 0.3 & 0.05 \\ 0.45 & 0.2 \end{pmatrix}$$
with $$X$$ indexing the columns and $$Y$$ the rows. Then (because the columns are constant multiples from one dist. to the other) $$P(Y\mid X)$$ is the same for the two, but $$P(X\mid Y)$$ is not. So this is an instance of covariate shift, but not label shift.